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5. Let f: X→Y be a function. (a) Show that A C {(f(A)) for any A E P(X). (b) Show that f(f(B)) C B for any BE P(Y). L4L

5. Let f: X→Y be a function. (a) Show that A C {(f(A)) for any A E P(X). (b) Show that f(f(B)) C B for any BE P(Y). L4L

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 fogoj3f. (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. 5. Let f: X→Y be a function. (a) Show that AC f(f(A)) for any A E P(X). (b) Show that f (f (B)) C B for any BE P(Y). 6. Let A, B be finite sets. Show that if |A \ B = |B\ A|, then |A| = |B
Transcribed Image Text:4. Let f: XY and g: Y X be functions such that fogoj3f. (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. 5. Let f: X→Y be a function. (a) Show that AC f(f(A)) for any A E P(X). (b) Show that f (f (B)) C B for any BE P(Y). 6. Let A, B be finite sets. Show that if |A \ B = |B\ A|, then |A| = |B
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