hw19 - f x = x 2-4 x 2 1 Show that f is a bijection and...

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Intro to Abstract Math Homework 19 Fall 2009 Due October 28 Exercise 56. Let f : X Y and g : Y Z be functions. a. Prove that if f and g are both surjective, then g f is surjective. b. Prove that if f and g are both bijective, then ( g f ) - 1 = f - 1 g - 1 . Hint: Compose the latter with g f and use the fact that inverses are unique. Exercise 57. a. Show that if x 0 then - 4 x 2 - 4 x 2 + 1 < 1. b. Let f : [0 , ) [ - 4 , 1) be deﬁned by
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Unformatted text preview: f ( x ) = x 2-4 x 2 + 1 . Show that f is a bijection and ﬁnd f-1 . Hint: If you can ﬁnd the inverse of f , this will prove f is a bijection. Exercise 58. Let S denote the collection of all sets. Prove that ≈ = { ( X,Y ) ∈ S 2 | there is a bijection f : X → Y } is an equivalence relation on S ....
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