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Question: o chape maa 4226 advanced calculus kercheval spring 2019 functions...

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o chape MAA 4226 Advanced Calculus- Kercheval Spring 2019 Functions, Sets, Cardinality Functions, Power Sets, Inverses If A and B are sets, a function /-A → B is a subset of the cartesian product A x B sich that subuets of A. A fanction f:ABinduces a function from 24 to 2 which we also call (even (a.b) E 1, (a and -a implies b V. The power set 24 of a set A is the collection of all thoagh it is a diferent function), by the rule fx) a) : z E X) for any X 24 Similarly there is a function 224 defined, for Y 2, by e power set of A is simply a collection of subsets of A, and can also considered an element of 224), We will use the notation uc(a EA:aC for some C e C e 21 We also have Notice also f(c) 22) and (D) E 22t) 1. Prove the following statements for f : A → B, X, X2 e 24, y, y, e 28CC 24, p C 21, (a) f(uc)-uf(c) (b) f(ne) cnf(c) (d) f-1 (nD)= nf-1(D) 2. Give examples showing that the the subset relation in item 1(b) cannot in general be replaced by equality. Then find conditions on f that would guarantee equality. Injections, Surjections, Bijections Recall that A-+ B is an injection if for all z, y E A, f(z)-f(y) İnnplies z-y, and f is a surjection if f(A)- B. The function f is a bijection if it is both an injection and a surjection. 3. Let f: AB, g:B C be functions. Prove: (a) if f and g are injective, then so is go f (b) if f and g are surjective, then so is go f

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