# Explicit Rule, Pair and Finite-Table Semantic Bridges

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261021-r01-mathematical-representation-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-3, b=-12, inputs=[-3, -2, 0]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-3, -12, [-3, -2, 0], [[-3, -3], [-2, -6], [0, -12]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-3, -12, [-3, -2, 0], [[-3, -3], [-2, -6], [0, -12]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0001/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 2. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=-11, inputs=[-2, -1, 1]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-2, -11, [-2, -1, 1], [[-2, -7], [-1, -9], [1, -13]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, -11, [-2, -1, 1], [[-2, -7], [-1, -9], [1, -13]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0002/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 3. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-10, inputs=[-1, 0, 2]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-1, -10, [-1, 0, 2], [[-1, -9], [0, -10], [2, -12]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, -10, [-1, 0, 2], [[-1, -9], [0, -10], [2, -12]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0003/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 4. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=-9, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(0, -9, [0, 1, 3], [[0, -9], [1, -9], [3, -9]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, -9, [0, 1, 3], [[0, -9], [1, -9], [3, -9]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0004/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 5. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=-8, inputs=[1, 2, 4]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(1, -8, [1, 2, 4], [[1, -7], [2, -6], [4, -4]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, -8, [1, 2, 4], [[1, -7], [2, -6], [4, -4]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0005/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 6. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=-7, inputs=[2, 3, 5]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(2, -7, [2, 3, 5], [[2, -3], [3, -1], [5, 3]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, -7, [2, 3, 5], [[2, -3], [3, -1], [5, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0006/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 7. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=3, b=-6, inputs=[3, 4, 6]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(3, -6, [3, 4, 6], [[3, 3], [4, 6], [6, 12]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, -6, [3, 4, 6], [[3, 3], [4, 6], [6, 12]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0007/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 8. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-3, b=-5, inputs=[-3, -2, 0]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-3, -5, [-3, -2, 0], [[-3, 4], [-2, 1], [0, -5]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-3, -5, [-3, -2, 0], [[-3, 4], [-2, 1], [0, -5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0008/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 9. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=-4, inputs=[-2, -1, 1]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-2, -4, [-2, -1, 1], [[-2, 0], [-1, -2], [1, -6]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, -4, [-2, -1, 1], [[-2, 0], [-1, -2], [1, -6]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0009/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 10. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-3, inputs=[-1, 0, 2]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-1, -3, [-1, 0, 2], [[-1, -2], [0, -3], [2, -5]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, -3, [-1, 0, 2], [[-1, -2], [0, -3], [2, -5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0010/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 11. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=-2, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(0, -2, [0, 1, 3], [[0, -2], [1, -2], [3, -2]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, -2, [0, 1, 3], [[0, -2], [1, -2], [3, -2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0011/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 12. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=-1, inputs=[1, 2, 4]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(1, -1, [1, 2, 4], [[1, 0], [2, 1], [4, 3]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, -1, [1, 2, 4], [[1, 0], [2, 1], [4, 3]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0012/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 13. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=0, inputs=[2, 3, 5]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(2, 0, [2, 3, 5], [[2, 4], [3, 6], [5, 10]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 0, [2, 3, 5], [[2, 4], [3, 6], [5, 10]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0013/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 14. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=3, b=1, inputs=[3, 4, 6]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(3, 1, [3, 4, 6], [[3, 10], [4, 13], [6, 19]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 1, [3, 4, 6], [[3, 10], [4, 13], [6, 19]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0014/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 15. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-3, b=2, inputs=[-3, -2, 0]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-3, 2, [-3, -2, 0], [[-3, 11], [-2, 8], [0, 2]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-3, 2, [-3, -2, 0], [[-3, 11], [-2, 8], [0, 2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0015/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 16. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=3, inputs=[-2, -1, 1]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [1, 1]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [1, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0016/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 17. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=4, inputs=[-1, 0, 2]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-1, 4, [-1, 0, 2], [[-1, 5], [0, 4], [2, 2]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 4, [-1, 0, 2], [[-1, 5], [0, 4], [2, 2]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0017/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 18. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=5, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(0, 5, [0, 1, 3], [[0, 5], [1, 5], [3, 5]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 5, [0, 1, 3], [[0, 5], [1, 5], [3, 5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0018/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 19. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=6, inputs=[1, 2, 4]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(1, 6, [1, 2, 4], [[1, 7], [2, 8], [4, 10]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 6, [1, 2, 4], [[1, 7], [2, 8], [4, 10]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0019/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 20. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=7, inputs=[2, 3, 5]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(2, 7, [2, 3, 5], [[2, 11], [3, 13], [5, 17]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 7, [2, 3, 5], [[2, 11], [3, 13], [5, 17]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0020/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 21. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=3, b=8, inputs=[3, 4, 6]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(3, 8, [3, 4, 6], [[3, 17], [4, 20], [6, 26]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 8, [3, 4, 6], [[3, 17], [4, 20], [6, 26]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0021/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 22. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-3, b=9, inputs=[-3, -2, 0]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-3, 9, [-3, -2, 0], [[-3, 18], [-2, 15], [0, 9]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-3, 9, [-3, -2, 0], [[-3, 18], [-2, 15], [0, 9]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0022/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 23. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=10, inputs=[-2, -1, 1]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-2, 10, [-2, -1, 1], [[-2, 14], [-1, 12], [1, 8]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 10, [-2, -1, 1], [[-2, 14], [-1, 12], [1, 8]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0023/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 24. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=11, inputs=[-1, 0, 2]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(-1, 11, [-1, 0, 2], [[-1, 12], [0, 11], [2, 9]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 11, [-1, 0, 2], [[-1, 12], [0, 11], [2, 9]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0024/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 25. A Rule-to-Table Bridge on an Explicit Finite Domain

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=12, inputs=[0, 1, 3]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).

View answer and explanation(0, 12, [0, 1, 3], [[0, 12], [1, 12], [3, 12]])

Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 12, [0, 1, 3], [[0, 12], [1, 12], [3, 12]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0025/)

Mathematical representation · qf-app21-representation-affine-table · candidate





## 26. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-2, -6], [-1, -9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, -9], [-2, -6]], -3, -12)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, -9], [-2, -6]], -3, -12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0026/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 27. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-1, -9], [1, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, -9], [1, -13]], -2, -11)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, -9], [1, -13]], -2, -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0027/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 28. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, -10], [3, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[0, -10], [3, -13]], -1, -10)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -10], [3, -13]], -1, -10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0028/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 29. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[1, -9], [2, -9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[1, -9], [2, -9]], 0, -9)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[1, -9], [2, -9]], 0, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0029/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 30. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, -6], [4, -4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[2, -6], [4, -4]], 1, -8)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[2, -6], [4, -4]], 1, -8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0030/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 31. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-2, -11], [1, -5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-2, -11], [1, -5]], 2, -7)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-2, -11], [1, -5]], 2, -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0031/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 32. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-1, -9], [0, -6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, -9], [0, -6]], 3, -6)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, -9], [0, -6]], 3, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0032/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 33. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, -5], [2, -11]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[0, -5], [2, -11]], -3, -5)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -5], [2, -11]], -3, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0033/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 34. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[1, -6], [4, -12]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[1, -6], [4, -12]], -2, -4)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[1, -6], [4, -12]], -2, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0034/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 35. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, -5], [3, -6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[2, -5], [3, -6]], -1, -3)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[2, -5], [3, -6]], -1, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0035/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 36. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-2, -2], [0, -2]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-2, -2], [0, -2]], 0, -2)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-2, -2], [0, -2]], 0, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0036/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 37. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-1, -2], [2, 1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, -2], [2, 1]], 1, -1)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, -2], [2, 1]], 1, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0037/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 38. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, 0], [1, 2]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[0, 0], [1, 2]], 2, 0)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 0], [1, 2]], 2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0038/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 39. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[1, 4], [3, 10]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[1, 4], [3, 10]], 3, 1)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[1, 4], [3, 10]], 3, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0039/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 40. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, -4], [5, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[2, -4], [5, -13]], -3, 2)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[2, -4], [5, -13]], -3, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0040/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 41. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-2, 7], [-1, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, 5], [-2, 7]], -2, 3)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 5], [-2, 7]], -2, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0041/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 42. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-1, 5], [1, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, 5], [1, 3]], -1, 4)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 5], [1, 3]], -1, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0042/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 43. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, 5], [3, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[0, 5], [3, 5]], 0, 5)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 5], [3, 5]], 0, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0043/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 44. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[1, 7], [2, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[1, 7], [2, 8]], 1, 6)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[1, 7], [2, 8]], 1, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0044/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 45. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, 11], [4, 15]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[2, 11], [4, 15]], 2, 7)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[2, 11], [4, 15]], 2, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0045/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 46. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-2, 2], [1, 11]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-2, 2], [1, 11]], 3, 8)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-2, 2], [1, 11]], 3, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0046/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 47. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[-1, 12], [0, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[-1, 12], [0, 9]], -3, 9)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[-1, 12], [0, 9]], -3, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0047/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 48. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[0, 10], [2, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[0, 10], [2, 6]], -2, 10)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 10], [2, 6]], -2, 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0048/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 49. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[1, 10], [4, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[1, 10], [4, 7]], -1, 11)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[1, 10], [4, 7]], -1, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0049/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 50. The Nonvertical Affine Rule through Two Distinct Inputs

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Points=[[2, 12], [3, 12]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Require exactly two distinct pairs with different x-coordinates. Give (canonical points,reduced rational slope,intercept), with no /1 for integer results.

View answer and explanation([[2, 12], [3, 12]], 0, 12)

Use the difference quotient for the slope and substitute either point to obtain the intercept; duplicate copies of a point do not create a new point. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[2, 12], [3, 12]], 0, 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0050/)

Mathematical representation · qf-app21-representation-two-point-rule · candidate





## 51. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -12], [1, -11], [1, -10], [2, -5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -12], [1, -10], [1, -11], [2, -5]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -12], [1, -10], [1, -11], [2, -5]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0051/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 52. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -11], [1, -10], [1, -10], [2, -3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -11], [1, -10], [2, -3]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -11], [1, -10], [2, -3]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0052/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 53. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -10], [1, -9], [1, -9], [2, -1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -10], [1, -9], [2, -1]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -10], [1, -9], [2, -1]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0053/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 54. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -9], [1, -8], [1, -8], [2, 1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -9], [1, -8], [2, 1]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -9], [1, -8], [2, 1]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0054/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 55. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -8], [1, -7], [1, -6], [2, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -8], [1, -6], [1, -7], [2, 3]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -8], [1, -6], [1, -7], [2, 3]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0055/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 56. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -7], [1, -6], [1, -6], [2, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -7], [1, -6], [2, 5]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -7], [1, -6], [2, 5]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0056/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 57. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -6], [1, -5], [1, -5], [2, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -6], [1, -5], [2, 7]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -6], [1, -5], [2, 7]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0057/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 58. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -5], [1, -4], [1, -4], [2, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -5], [1, -4], [2, 9]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -5], [1, -4], [2, 9]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0058/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 59. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -4], [1, -3], [1, -2], [2, 11]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -4], [1, -2], [1, -3], [2, 11]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -4], [1, -2], [1, -3], [2, 11]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0059/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 60. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -3], [1, -2], [1, -2], [2, 13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -3], [1, -2], [2, 13]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -3], [1, -2], [2, 13]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0060/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 61. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -2], [1, -1], [1, -1], [2, 15]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -2], [1, -1], [2, 15]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -2], [1, -1], [2, 15]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0061/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 62. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, -1], [1, 0], [1, 0], [2, 17]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, -1], [1, 0], [2, 17]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, -1], [1, 0], [2, 17]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0062/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 63. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 0], [1, 1], [1, 2], [2, 19]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 0], [1, 1], [1, 2], [2, 19]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 0], [1, 1], [1, 2], [2, 19]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0063/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 64. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 1], [1, 2], [1, 2], [2, 21]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 1], [1, 2], [2, 21]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 1], [1, 2], [2, 21]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0064/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 65. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 2], [1, 3], [1, 3], [2, 23]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 2], [1, 3], [2, 23]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 2], [1, 3], [2, 23]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0065/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 66. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 3], [1, 4], [1, 4], [2, 25]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 3], [1, 4], [2, 25]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 3], [1, 4], [2, 25]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0066/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 67. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 4], [1, 5], [1, 6], [2, 27]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 4], [1, 5], [1, 6], [2, 27]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 4], [1, 5], [1, 6], [2, 27]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0067/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 68. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 5], [1, 6], [1, 6], [2, 29]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 5], [1, 6], [2, 29]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 5], [1, 6], [2, 29]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0068/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 69. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 6], [1, 7], [1, 7], [2, 31]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 6], [1, 7], [2, 31]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 6], [1, 7], [2, 31]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0069/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 70. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 7], [1, 8], [1, 8], [2, 33]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 7], [1, 8], [2, 33]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 7], [1, 8], [2, 33]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0070/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 71. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 8], [1, 9], [1, 10], [2, 35]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 8], [1, 10], [1, 9], [2, 35]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 8], [1, 10], [1, 9], [2, 35]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0071/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 72. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 9], [1, 10], [1, 10], [2, 37]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 9], [1, 10], [2, 37]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 9], [1, 10], [2, 37]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0072/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 73. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 10], [1, 11], [1, 11], [2, 39]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 10], [1, 11], [2, 39]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 10], [1, 11], [2, 39]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0073/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 74. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 11], [1, 12], [1, 12], [2, 41]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 11], [1, 12], [2, 41]], [0, 1, 2], [], Yes)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 11], [1, 12], [2, 41]], [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0074/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 75. Repeated Inputs versus a Single-Valued Pair Table

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. Pairs=[[0, 12], [1, 13], [1, 14], [2, 43]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (canonical pairs,sorted distinct domain,inputs having multiple distinct outputs,is the table a function on its domain?). List conflicts in first-appearance order in the canonical pair table.

View answer and explanation([[0, 12], [1, 13], [1, 14], [2, 43]], [0, 1, 2], [1], No)

Identical duplicate pairs are harmless; one input with distinct outputs violates the function condition. Domain order and canonical pair order are separate. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([[0, 12], [1, 13], [1, 14], [2, 43]], [0, 1, 2], [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0075/)

Mathematical representation · qf-app21-representation-function-audit · candidate





## 76. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=-10, pairs=[[0, -10], [1, -12], [2, -13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-2, -10, [[0, -10, -10, Yes], [1, -12, -12, Yes], [2, -13, -14, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, -10, [[0, -10, -10, Yes], [1, -12, -12, Yes], [2, -13, -14, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0076/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 77. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-9, pairs=[[0, -9], [1, -10], [2, -11]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-1, -9, [[0, -9, -9, Yes], [1, -10, -10, Yes], [2, -11, -11, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, -9, [[0, -9, -9, Yes], [1, -10, -10, Yes], [2, -11, -11, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0077/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 78. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=-8, pairs=[[0, -8], [1, -8], [2, -8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(0, -8, [[0, -8, -8, Yes], [1, -8, -8, Yes], [2, -8, -8, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, -8, [[0, -8, -8, Yes], [1, -8, -8, Yes], [2, -8, -8, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0078/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 79. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=-7, pairs=[[0, -7], [1, -6], [2, -5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(1, -7, [[0, -7, -7, Yes], [1, -6, -6, Yes], [2, -5, -5, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, -7, [[0, -7, -7, Yes], [1, -6, -6, Yes], [2, -5, -5, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0079/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 80. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=-6, pairs=[[0, -6], [1, -4], [2, -1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(2, -6, [[0, -6, -6, Yes], [1, -4, -4, Yes], [2, -1, -2, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, -6, [[0, -6, -6, Yes], [1, -4, -4, Yes], [2, -1, -2, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0080/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 81. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=-5, pairs=[[0, -5], [1, -7], [2, -9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-2, -5, [[0, -5, -5, Yes], [1, -7, -7, Yes], [2, -9, -9, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, -5, [[0, -5, -5, Yes], [1, -7, -7, Yes], [2, -9, -9, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0081/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 82. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-4, pairs=[[0, -4], [1, -5], [2, -6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-1, -4, [[0, -4, -4, Yes], [1, -5, -5, Yes], [2, -6, -6, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, -4, [[0, -4, -4, Yes], [1, -5, -5, Yes], [2, -6, -6, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0082/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 83. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=-3, pairs=[[0, -3], [1, -3], [2, -3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(0, -3, [[0, -3, -3, Yes], [1, -3, -3, Yes], [2, -3, -3, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, -3, [[0, -3, -3, Yes], [1, -3, -3, Yes], [2, -3, -3, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0083/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 84. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=-2, pairs=[[0, -2], [1, -1], [2, 1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(1, -2, [[0, -2, -2, Yes], [1, -1, -1, Yes], [2, 1, 0, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, -2, [[0, -2, -2, Yes], [1, -1, -1, Yes], [2, 1, 0, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0084/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 85. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=-1, pairs=[[0, -1], [1, 1], [2, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(2, -1, [[0, -1, -1, Yes], [1, 1, 1, Yes], [2, 3, 3, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, -1, [[0, -1, -1, Yes], [1, 1, 1, Yes], [2, 3, 3, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0085/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 86. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=0, pairs=[[0, 0], [1, -2], [2, -4]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-2, 0, [[0, 0, 0, Yes], [1, -2, -2, Yes], [2, -4, -4, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 0, [[0, 0, 0, Yes], [1, -2, -2, Yes], [2, -4, -4, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0086/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 87. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=1, pairs=[[0, 1], [1, 0], [2, -1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-1, 1, [[0, 1, 1, Yes], [1, 0, 0, Yes], [2, -1, -1, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 1, [[0, 1, 1, Yes], [1, 0, 0, Yes], [2, -1, -1, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0087/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 88. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=2, pairs=[[0, 2], [1, 2], [2, 3]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(0, 2, [[0, 2, 2, Yes], [1, 2, 2, Yes], [2, 3, 2, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 2, [[0, 2, 2, Yes], [1, 2, 2, Yes], [2, 3, 2, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0088/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 89. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=3, pairs=[[0, 3], [1, 4], [2, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(1, 3, [[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 3, [[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 5, 5, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0089/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 90. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=4, pairs=[[0, 4], [1, 6], [2, 8]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(2, 4, [[0, 4, 4, Yes], [1, 6, 6, Yes], [2, 8, 8, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 4, [[0, 4, 4, Yes], [1, 6, 6, Yes], [2, 8, 8, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0090/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 91. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=5, pairs=[[0, 5], [1, 3], [2, 1]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-2, 5, [[0, 5, 5, Yes], [1, 3, 3, Yes], [2, 1, 1, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 5, [[0, 5, 5, Yes], [1, 3, 3, Yes], [2, 1, 1, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0091/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 92. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=6, pairs=[[0, 6], [1, 5], [2, 5]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-1, 6, [[0, 6, 6, Yes], [1, 5, 5, Yes], [2, 5, 4, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 6, [[0, 6, 6, Yes], [1, 5, 5, Yes], [2, 5, 4, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0092/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 93. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=7, pairs=[[0, 7], [1, 7], [2, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(0, 7, [[0, 7, 7, Yes], [1, 7, 7, Yes], [2, 7, 7, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 7, [[0, 7, 7, Yes], [1, 7, 7, Yes], [2, 7, 7, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0093/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 94. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=8, pairs=[[0, 8], [1, 9], [2, 10]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(1, 8, [[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 8, [[0, 8, 8, Yes], [1, 9, 9, Yes], [2, 10, 10, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0094/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 95. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=9, pairs=[[0, 9], [1, 11], [2, 13]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(2, 9, [[0, 9, 9, Yes], [1, 11, 11, Yes], [2, 13, 13, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 9, [[0, 9, 9, Yes], [1, 11, 11, Yes], [2, 13, 13, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0095/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 96. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=10, pairs=[[0, 10], [1, 8], [2, 7]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-2, 10, [[0, 10, 10, Yes], [1, 8, 8, Yes], [2, 7, 6, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 10, [[0, 10, 10, Yes], [1, 8, 8, Yes], [2, 7, 6, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0096/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 97. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=11, pairs=[[0, 11], [1, 10], [2, 9]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(-1, 11, [[0, 11, 11, Yes], [1, 10, 10, Yes], [2, 9, 9, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 11, [[0, 11, 11, Yes], [1, 10, 10, Yes], [2, 9, 9, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0097/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 98. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=0, b=12, pairs=[[0, 12], [1, 12], [2, 12]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(0, 12, [[0, 12, 12, Yes], [1, 12, 12, Yes], [2, 12, 12, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 12, [[0, 12, 12, Yes], [1, 12, 12, Yes], [2, 12, 12, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0098/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 99. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=1, b=13, pairs=[[0, 13], [1, 14], [2, 15]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(1, 13, [[0, 13, 13, Yes], [1, 14, 14, Yes], [2, 15, 15, Yes]], Yes)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 13, [[0, 13, 13, Yes], [1, 14, 14, Yes], [2, 15, 15, Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0099/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate





## 100. Audit Every Table Row against an Affine Rule

Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=2, b=14, pairs=[[0, 14], [1, 16], [2, 19]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).

View answer and explanation(2, 14, [[0, 14, 14, Yes], [1, 16, 16, Yes], [2, 19, 18, No]], No)

Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 14, [[0, 14, 14, Yes], [1, 16, 16, Yes], [2, 19, 18, No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261021-r01-mathematical-representation-0100/)

Mathematical representation · qf-app21-representation-consistency-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/validation-report.json)