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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261004-advance-20261005-r01-injective-foundations · 2026-10-04 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
The declared total function f:A→B has A=∅, B=∅, and table ∅. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
Yes
Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.
No
Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-table-property · candidate
The declared total function f:A→B has A=∅, B=∅, and table ∅. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-1, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-1, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-1, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-1, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-2, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-2, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-2, -1)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-2, -1); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-3, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-3, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-3, -2)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-3, -2); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-4, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-4, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-4, -3)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-4, -3); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-5, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-5, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-5, -4)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-5, -4); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
none
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-6, 0)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-6, 0); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.
(-6, -5)
Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-6, -5); repeated copies of the same pair are not distinct inputs.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-first-collision · candidate
The declared total function f:A→B has A=∅, B={0}, and table ∅. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 0. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. For target t=1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 1. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. For target t=5∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 5. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
2
Enumerate inputs in A whose output is exactly 0. The fiber cardinality is 2; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. For target t=-2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly -2. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. For target t=2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 2. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. For target t=7∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 7. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. For target t=1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 1. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. For target t=-4∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly -4. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. For target t=1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 1. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. For target t=9∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 9. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. For target t=2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 2. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. For target t=-3∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly -3. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For target t=3∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 3. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. For target t=11∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 11. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. For target t=3∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 3. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For target t=-2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
2
Enumerate inputs in A whose output is exactly -2. The fiber cardinality is 2; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For target t=5∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 5. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. For target t=13∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 13. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For target t=4∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 4. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. For target t=-1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
2
Enumerate inputs in A whose output is exactly -1. The fiber cardinality is 2; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 0. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. For target t=15∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly 15. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For target t=5∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
1
Enumerate inputs in A whose output is exactly 5. The fiber cardinality is 1; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. For target t=-8∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.
0
Enumerate inputs in A whose output is exactly -8. The fiber cardinality is 0; it is neither an output value nor a probability.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate
The declared total function f:A→B has A={0, 1}, B={0}, and table {(0, 0), (1, 0)}. For C=∅⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. For C={0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. For C={0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
No
Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. For C={0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. For C={-2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. For C={-2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. For C={-2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
No
Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
No
Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
No
Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. For C={-5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
No
Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. For C={-6, -5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. For C={-5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For C={-5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
Yes
Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. For C={-6, -5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.
No
Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-injective-restriction-property · candidate
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