Explicit Rule, Pair and Finite-Table Semantic Bridges
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Mathematical representation · qf-app21-representation-consistency-audit · candidate
Canonical question JSONL · English JSONL · Validation report