hw17 - g P A → P A given by g X = A-X We have already...

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Intro to Abstract Math Homework 17 Fall 2009 Due October 19 Exercise 47. Let A = { 1 , 2 , 3 } and B = {♦ , ♥} . Show that there is no injection f : A B . Exercise 48. Let f : X Y be a function. a. If A i X for i I , prove that f \ i I A i ! \ i I f ( A i ) . Find an example to show that these two sets need not be equal. b. If B i Y for i I , prove that f - 1 \ i I B i ! = \ i I f - 1 ( B i ) . Exercise 49. Show that h : [ - 3 , ) [1 , ) given by h ( x ) = 1 + x + 3 is a bijection. Exercise 50. Let A be a set. Recall the function
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Unformatted text preview: g : P ( A ) → P ( A ) given by g ( X ) = A-X . We have already seen that this function is a surjection. Prove that, in fact, it is a bijection. Exercise 51. Let A be a non-empty set and let f : A → P ( A ) be any function. Show that f is not surjective. Hint: Consider S = { a ∈ A | a 6∈ f ( a ) } ∈ P ( A )....
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