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4. Let f: XY and g: Y X be functions such that fogof=f. (a) Prove that f is surjective if and only if f og = idy. (b) Prove that f is injective if and only if gof = idx. (c) Prove that if f is injective, then g is surjective. %3D (d) Prove that if f is surjective, then g is injective.

4. Let f: XY and g: Y X be functions such that fogof=f. (a) Prove that f is surjective if and only if f og = idy. (b) Prove that f is injective if and only if gof = idx. (c) Prove that if f is injective, then g is surjective. %3D (d) Prove that if f is surjective, then g is injective.

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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4. Let f: XY and g: Y X be functions such that fogof =f. (a) Prove that f is surjective if and only if fog =idy. (b) Prove that f is injective if and only if gof = idx. (c) Prove that if f is injective, then g is surjective. (d) Prove that if f is surjective, then g is injective. Solution: (a) First, suppose that f is surjective. Let y E Y. Since f there exists x EX such that f(x) = y. Since fogof= f, we have f(9(y)) = f(g(f(x)) = f(x) = y. It follows that fog=idy. Now, suppose that fog=idy. Let y E Y be given, and set x = g(y). Then f(x) = f(g(y)) = idy (y) = y. It follows that f is surjective. (c) Suppose that f is injective. Let x EX be given, and set y = f(x). E assumption that fogof= f, we have f (x) = f(q(f(x))). PS 3 Fall 2021 (1).pdf Horizontal Mode...xlsx MacBookPro Search or type URL ¥ $ .RB 3 4 7 9. LO
Transcribed Image Text:4. Let f: XY and g: Y X be functions such that fogof =f. (a) Prove that f is surjective if and only if fog =idy. (b) Prove that f is injective if and only if gof = idx. (c) Prove that if f is injective, then g is surjective. (d) Prove that if f is surjective, then g is injective. Solution: (a) First, suppose that f is surjective. Let y E Y. Since f there exists x EX such that f(x) = y. Since fogof= f, we have f(9(y)) = f(g(f(x)) = f(x) = y. It follows that fog=idy. Now, suppose that fog=idy. Let y E Y be given, and set x = g(y). Then f(x) = f(g(y)) = idy (y) = y. It follows that f is surjective. (c) Suppose that f is injective. Let x EX be given, and set y = f(x). E assumption that fogof= f, we have f (x) = f(q(f(x))). PS 3 Fall 2021 (1).pdf Horizontal Mode...xlsx MacBookPro Search or type URL ¥ $ .RB 3 4 7 9. LO
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