OCMEBilingual mathematics & derivations繁體中文

Typed Composition Paths and Slope-First Affine Orders

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batch-20261011-advance-20261015-r04-composition-foundations · 2026-10-11 · 100 questions

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

1. Complete Compatible Composition Paths and Range

First f, then g: Fin(0)=∅→Fin(0)→Fin(0); complete f=[], g=[]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([0, 0, 0], [], [])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([0, 0, 0], [], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

2. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 0, 3]], [3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 0, 3]], [3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

3. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 0, 3]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 0, 3]], [1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

4. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 0, 3]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 0, 3]], [0, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

5. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 0, 3]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 0, 3]], [1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

6. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 0, 3]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 0, 3]], [1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

7. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 0, 3]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 0, 3]], [0, 1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

8. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 0, 3]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 0, 3]], [0, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

9. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 0, 3]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 0, 3]], [0, 1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

10. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 2, 0], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 0, 3]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 0, 3]], [0, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

11. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 1, 1]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 1, 1]], [1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

12. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 1, 1]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 1, 1]], [1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

13. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 1, 1]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 1, 1]], [0, 1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

14. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 1, 1]], [1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 1, 1]], [1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

15. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 1, 1]], [1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 1, 1]], [1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

16. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 1, 1]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 1, 1]], [0, 1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

17. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 1, 1]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 2, 0], [2, 1, 1]], [0, 1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

18. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 1, 1]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 2, 0], [2, 1, 1]], [0, 1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

19. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 2, 1], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 1, 1]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 2, 0], [2, 1, 1]], [0, 1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

20. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 2, 0]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 0, 3], [2, 2, 0]], [0, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

21. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 2, 0]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 0, 3], [2, 2, 0]], [0, 1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

22. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 0, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 2, 0]], [0, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 0, 3], [2, 2, 0]], [0, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

23. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[0, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 2, 0]], [0, 1, 3])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 0, 3], [1, 1, 1], [2, 2, 0]], [0, 1, 3]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

24. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[1, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 2, 0]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 1, 1], [1, 1, 1], [2, 2, 0]], [0, 1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

25. Complete Compatible Composition Paths and Range

First f, then g: Fin(3)={0,…,2}→Fin(3)→Fin(4); complete f=[2, 1, 2], g=[3, 1, 0]. Give ([n,m,k], every [x,f(x),g(f(x))] row, increasing composition range). Unused cells must still be typed.

View answer and explanation

([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 2, 0]], [0, 1])

Read f before g at every source point, retain each intermediate value, and deduplicate final outputs; composition is not multiplication or arbitrary exchange. The exact result, retaining the requested order, is ([3, 3, 4], [[0, 2, 0], [1, 1, 1], [2, 2, 0]], [0, 1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-compatible-path-table · candidate

26. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(0)=∅, middle=Fin(0)=∅, target=Fin(0)=∅. Complete f=[]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([0, 0, 0], [], [], [], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([0, 0, 0], [], [], [], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

27. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(3)={0,…,2}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 1, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([3, 3, 3], [0, 2], [0, 2], [2, 1], [1])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([3, 3, 3], [0, 2], [0, 2], [2, 1], [1]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

28. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(3)={0,…,2}, middle=Fin(3)={0,…,2}, target=Fin(0)=∅. Complete f=[0, 1, 2]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([3, 3, 0], [], [], [], [0, 1, 2])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([3, 3, 0], [], [], [], [0, 1, 2]).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

29. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

30. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

31. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

32. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

33. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

34. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

35. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

36. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 2, 0, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 2, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 2, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

37. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

38. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

39. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

40. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

41. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

42. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

43. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 1, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

44. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 2, 2, 0]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 1, 2], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 1, 1, 2], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

45. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 2, 2, 1], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

46. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 2, 2, 1], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

47. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 2, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 1, 2, 2, 1], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

48. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 2, 0, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 1, 2, 2, 1], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

49. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[0, 0, 2, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [2, 2, 1, 2, 1], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

50. Restricted Middle Domain with Original Pairing Preserved

First f, then g: Fin(5)={0,…,4}, middle=Fin(3)={0,…,2}, target=Fin(3)={0,…,2}. Complete f=[2, 0, 2, 0, 2]; g has original ordered dom=[2, 0] paired with g values=[1, 2]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.

View answer and explanation

([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 1], [])

Construct g’s lookup using its original pairing before sorting the displayed domain; evaluate only when f(x) lies in that domain. Never evaluate g outside it. The exact result, retaining the requested order, is ([5, 3, 3], [0, 2], [0, 1, 2, 3, 4], [1, 2, 1, 2, 1], []).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-restricted-table-domain · candidate

51. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(1)x+(0), g(x)=(1)x+(0); ordered inputs=[0/1, 1/1, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([1, 0], [1, 0], [0/1, 1/1, -1/1], [0/1, 1/1, -1/1], 0)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([1, 0], [1, 0], [0/1, 1/1, -1/1], [0/1, 1/1, -1/1], 0).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

52. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(1)x+(1), g(x)=(2)x+(0); ordered inputs=[0/1, 1/1, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([2, 2], [2, 1], [2/1, 4/1, 0/1], [1/1, 3/1, -1/1], 1)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([2, 2], [2, 1], [2/1, 4/1, 0/1], [1/1, 3/1, -1/1], 1).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

53. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(0)x+(1), g(x)=(0)x+(2); ordered inputs=[0/1, 1/1, 2/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([0, 2], [0, 1], [2/1, 2/1, 2/1], [1/1, 1/1, 1/1], 1)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([0, 2], [0, 1], [2/1, 2/1, 2/1], [1/1, 1/1, 1/1], 1).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

54. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(2)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-4, 3], [-4, 1], [3/1, 1/1, 7/1], [1/1, -1/1, 5/1], 2)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-4, 3], [-4, 1], [3/1, 1/1, 7/1], [1/1, -1/1, 5/1], 2).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

55. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(3)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-6, 1], [-6, 3], [1/1, -2/1, 7/1], [3/1, 0/1, 9/1], -2)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-6, 1], [-6, 3], [1/1, -2/1, 7/1], [3/1, 0/1, 9/1], -2).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

56. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(4)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-8, -1], [-8, 5], [-1/1, -5/1, 7/1], [5/1, 1/1, 13/1], -6)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-8, -1], [-8, 5], [-1/1, -5/1, 7/1], [5/1, 1/1, 13/1], -6).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

57. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(5)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-10, 3], [-10, 4], [3/1, -2/1, 13/1], [4/1, -1/1, 14/1], -1)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-10, 3], [-10, 4], [3/1, -2/1, 13/1], [4/1, -1/1, 14/1], -1).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

58. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(6)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-12, 1], [-12, 6], [1/1, -5/1, 13/1], [6/1, 0/1, 18/1], -5)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-12, 1], [-12, 6], [1/1, -5/1, 13/1], [6/1, 0/1, 18/1], -5).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

59. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(7)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-14, -1], [-14, 8], [-1/1, -8/1, 13/1], [8/1, 1/1, 22/1], -9)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-14, -1], [-14, 8], [-1/1, -8/1, 13/1], [8/1, 1/1, 22/1], -9).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

60. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(8)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-16, 3], [-16, 7], [3/1, -5/1, 19/1], [7/1, -1/1, 23/1], -4)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-16, 3], [-16, 7], [3/1, -5/1, 19/1], [7/1, -1/1, 23/1], -4).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

61. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(9)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

62. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(10)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-20, -1], [-20, 11], [-1/1, -11/1, 19/1], [11/1, 1/1, 31/1], -12)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-20, -1], [-20, 11], [-1/1, -11/1, 19/1], [11/1, 1/1, 31/1], -12).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

63. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(11)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-22, 3], [-22, 10], [3/1, -8/1, 25/1], [10/1, -1/1, 32/1], -7)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-22, 3], [-22, 10], [3/1, -8/1, 25/1], [10/1, -1/1, 32/1], -7).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

64. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(12)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-24, 1], [-24, 12], [1/1, -11/1, 25/1], [12/1, 0/1, 36/1], -11)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-24, 1], [-24, 12], [1/1, -11/1, 25/1], [12/1, 0/1, 36/1], -11).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

65. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(13)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-26, -1], [-26, 14], [-1/1, -14/1, 25/1], [14/1, 1/1, 40/1], -15).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

66. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(14)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-28, 3], [-28, 13], [3/1, -11/1, 31/1], [13/1, -1/1, 41/1], -10)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-28, 3], [-28, 13], [3/1, -11/1, 31/1], [13/1, -1/1, 41/1], -10).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

67. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(15)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-30, 1], [-30, 15], [1/1, -14/1, 31/1], [15/1, 0/1, 45/1], -14)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-30, 1], [-30, 15], [1/1, -14/1, 31/1], [15/1, 0/1, 45/1], -14).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

68. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(16)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-32, -1], [-32, 17], [-1/1, -17/1, 31/1], [17/1, 1/1, 49/1], -18)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-32, -1], [-32, 17], [-1/1, -17/1, 31/1], [17/1, 1/1, 49/1], -18).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

69. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(17)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-34, 3], [-34, 16], [3/1, -14/1, 37/1], [16/1, -1/1, 50/1], -13)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-34, 3], [-34, 16], [3/1, -14/1, 37/1], [16/1, -1/1, 50/1], -13).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

70. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(18)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-36, 1], [-36, 18], [1/1, -17/1, 37/1], [18/1, 0/1, 54/1], -17)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-36, 1], [-36, 18], [1/1, -17/1, 37/1], [18/1, 0/1, 54/1], -17).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

71. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(19)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-38, -1], [-38, 20], [-1/1, -20/1, 37/1], [20/1, 1/1, 58/1], -21)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-38, -1], [-38, 20], [-1/1, -20/1, 37/1], [20/1, 1/1, 58/1], -21).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

72. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(20)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-40, 3], [-40, 19], [3/1, -17/1, 43/1], [19/1, -1/1, 59/1], -16)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-40, 3], [-40, 19], [3/1, -17/1, 43/1], [19/1, -1/1, 59/1], -16).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

73. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(21)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-42, 1], [-42, 21], [1/1, -20/1, 43/1], [21/1, 0/1, 63/1], -20)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-42, 1], [-42, 21], [1/1, -20/1, 43/1], [21/1, 0/1, 63/1], -20).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

74. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(22)x+(1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-44, -1], [-44, 23], [-1/1, -23/1, 43/1], [23/1, 1/1, 67/1], -24)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-44, -1], [-44, 23], [-1/1, -23/1, 43/1], [23/1, 1/1, 67/1], -24).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

75. Both Affine Composition Orders with Slope-First Coefficients

Total Q functions f(x)=(23)x+(-1), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant).

View answer and explanation

([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19)

Substitute twice in each actual composition order; the coefficient pairs are explicitly slope-first, not ascending powers. The input-1 versus input-0 difference checks each slope; do not exchange evaluation order. The exact result, retaining the requested order, is ([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-noncommuting-affine · candidate

76. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(0)=∅→Fin(0)→Fin(0)→Fin(0); complete f=[], g=[], h=[]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([0, 0, 0, 0], [], [], [], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([0, 0, 0, 0], [], [], [], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

77. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 0, 0], [0, 0, 0], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 0], [0, 0, 0], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

78. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 0, 0], [1, 0, 0], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 0], [1, 0, 0], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

79. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 0, 0], [3, 0, 0], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 0], [3, 0, 0], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

80. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 1, 0], [0, 1, 0], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 0], [0, 1, 0], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

81. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 1, 0], [1, 1, 0], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 0], [1, 1, 0], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

82. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 1, 0], [3, 1, 0], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 0], [3, 1, 0], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

83. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 3, 0], [0, 3, 0], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 3, 0], [0, 3, 0], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

84. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 3, 0], [1, 3, 0], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 3, 0], [1, 3, 0], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

85. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 2, 0], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 3, 0], [3, 3, 0], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 0, 2, 0]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 3, 0], [3, 3, 0], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 0, 2, 0]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

86. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 0, 1], [0, 0, 1], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 1], [0, 0, 1], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

87. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 0, 1], [1, 0, 1], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 1], [1, 0, 1], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

88. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 0, 1], [3, 0, 1], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 1], [3, 0, 1], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

89. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 1, 1], [0, 1, 1], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 1], [0, 1, 1], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

90. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 1, 1], [1, 1, 1], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 1], [1, 1, 1], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

91. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 1, 1], [3, 1, 1], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 1], [3, 1, 1], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

92. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 3, 1], [0, 3, 1], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 3, 1], [0, 3, 1], [[0, 0, 2, 0], [1, 2, 1, 3], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

93. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 3, 1], [1, 3, 1], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 3, 1], [1, 3, 1], [[0, 1, 0, 1], [1, 2, 1, 3], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

94. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 2, 1], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 3, 1], [3, 3, 1], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 1, 0, 1]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 3, 1], [3, 3, 1], [[0, 2, 1, 3], [1, 2, 1, 3], [2, 1, 0, 1]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

95. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 0, 3], [0, 0, 3], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 0, 3], [0, 0, 3], [[0, 0, 2, 0], [1, 0, 2, 0], [2, 2, 1, 3]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

96. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 0, 3], [1, 0, 3], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 0, 3], [1, 0, 3], [[0, 1, 0, 1], [1, 0, 2, 0], [2, 2, 1, 3]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

97. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 0, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 0, 3], [3, 0, 3], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 0, 3], [3, 0, 3], [[0, 2, 1, 3], [1, 0, 2, 0], [2, 2, 1, 3]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

98. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[0, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [0, 1, 3], [0, 1, 3], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [0, 1, 3], [0, 1, 3], [[0, 0, 2, 0], [1, 1, 0, 1], [2, 2, 1, 3]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

99. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[1, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [1, 1, 3], [1, 1, 3], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [1, 1, 3], [1, 1, 3], [[0, 1, 0, 1], [1, 1, 0, 1], [2, 2, 1, 3]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

100. Two Parenthesizations of Three Complete Typed Maps

First f, then g, then h: Fin(3)={0,…,2}→Fin(3)→Fin(3)→Fin(4); complete f=[2, 1, 2], g=[2, 0, 1], h=[1, 3, 0]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.

View answer and explanation

([3, 3, 3, 4], [3, 1, 3], [3, 1, 3], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 2, 1, 3]], Yes)

Compute both parenthesized finite routes completely and compare their sourcewise values; this finite check is not a formal proof of every function identity. The exact result, retaining the requested order, is ([3, 3, 3, 4], [3, 1, 3], [3, 1, 3], [[0, 2, 1, 3], [1, 1, 0, 1], [2, 2, 1, 3]], Yes).

Reviewed step-by-step derivation

Function composition · qf-density15-composition-three-map-parentheses · candidate

Canonical question JSONL · English JSONL · Validation report