notes915 - Charley Crissman Math 55 Discussion Notes...

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Unformatted text preview: Charley Crissman Math 55 Discussion Notes September 15th, 2010 Injectivity and Surjectivity How to Prove Injectivity Suppose we want to show that the function h : A → B is injective. The proof always begins the same way: Proof. Let a 1 ,a 2 ∈ A be arbitrary, and assume that h ( a 1 ) = h ( a 2 ). We want to show that a 1 = a 2 ... An example: Example 0.1. Let f : N → Z be the function defined by f ( n ) = n 2- 5. I claim that f is injective. Proof. Let a,b ∈ N be arbitrary, and assume that f ( a ) = f ( b ). [Note how I don’t care what variable names I choose! Don’t get attached to letters!] We want to show that a = b . We have a 2- 5 = b 2- 5 , hence a 2 = b 2 . Taking square roots, we have | a | = | b | . Since a,b ∈ N , | a | = a and | b | = b , so a = b . How to Prove Surjectivity Suppose we want to show that the function k : C → D is surjective. The proof always begins the same way: Proof. Let d ∈ D be arbitrary. We want to find a c ∈ C such that k ( c ) = d .......
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notes915 - Charley Crissman Math 55 Discussion Notes...

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