H For any set C for any \u03d5\u03c8 Map CA if f \u03d5 f \u03c8 then\u03d5 \u03c8 4 Any functionf A B

# H for any set c for any ϕψ map ca if f ϕ f ψ

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(H) For any set C , for any ϕ,ψ Map ( C,A ), if f ϕ = f ψ then ϕ = ψ . 4. Any function f : A −→ B induces a pair of functions f P : P ( A ) −→ P ( B ), f P : P ( B ) −→ P ( A ), given by f P ( S )= f ( S ) for any S P ( A ), f P = f - 1 ( U ) for any U P ( B ) respectively. (a) Let f : A −→ B , g : A −→ B be functions. Prove that the following statements are equivalent: (A) f = g (B) f P = g P (C) f P = g P (b) Let f : A −→ B , g : B −→ C be functions. Prove the following statements: i. ( g f ) P = g P f P ii. ( g f ) P = f P g P (c) Let f : A −→ B be a function. Prove that the following statements are equivalent: (D) f is injective. (E) f P is injective. (F) f P is surjective. 1
5. Let A,B,C,D be non-empty sets. Suppose there is a bijective function from A to C and there is a bijective function from B to D . Prove the following statements: (a) There is a bijective function from A × B to C × D . (b) There is a bijective function from Map ( A,B ) to Map ( C,D ). 6. Prove the following statements: (a) Let A,B, be sets and ( A α ) α I be a non-empty family of subsets of A . parenleftBigg intersectiondisplay α I A α parenrightBigg B = intersectiondisplay α I ( A α B ) . (b) Let A,B, be sets and ( A α ) α I be a non-empty family of subsets of A .

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• Spring '20
• Inverse function, Finite set, Basic concepts in set theory, FP

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