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Suppose that A and B are non-empty sets, and f: A → B is a function. (a) Prove that fis injective if and only if there exists a function g: B → A such that g(f(x)) = x for all A. (b) Prove that fis surjective if and only if there exist a function h: B→ A such that f(h(y)) = y for all y = B. (c) Prove that if f is both injective and surjective, then there exists a unique function g: B→ A such that g(f(x)) : x for all x A and f(g(y)) y for all y = B.

Suppose that A and B are non-empty sets, and f: A → B is a function. (a) Prove that fis injective if and only if there exists a function g: B → A such that g(f(x)) = x for all A. (b) Prove that fis surjective if and only if there exist a function h: B→ A such that f(h(y)) = y for all y = B. (c) Prove that if f is both injective and surjective, then there exists a unique function g: B→ A such that g(f(x)) : x for all x A and f(g(y)) y for all y = B.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Suppose that A and B are non-empty sets, and f: A → B is a function. (a) Prove that fis injective if and only if there exists a function g: B → A such that g(f(x)) = x for all A. (b) Prove that fis surjective if and only if there exist a function h: B→ A such that f(h(y)) = y for all y = B. (c) Prove that if f is both injective and surjective, then there exists a unique function g: B→ A such that g(f(x)) : x for all x A and f(g(y)) y for all y = B.
Transcribed Image Text:Suppose that A and B are non-empty sets, and f: A → B is a function. (a) Prove that fis injective if and only if there exists a function g: B → A such that g(f(x)) = x for all A. (b) Prove that fis surjective if and only if there exist a function h: B→ A such that f(h(y)) = y for all y = B. (c) Prove that if f is both injective and surjective, then there exists a unique function g: B→ A such that g(f(x)) : x for all x A and f(g(y)) y for all y = B.
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