Proofs and Congruence

Proofs and Congruence - Math 100 Evens and Odds, First...

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Unformatted text preview: Math 100 Evens and Odds, First Proofs Martin H. Weissman To practice proof-writing, in a paragraph style, we will prove some basic facts about arithmetic. 1. Evens and Odds In order to prove things about even and odd numbers, we begin by recalling the definitions of even and odd. Definition 1. A number n is called even if ∃ a ∈ Z such that 2 a = n . Definition 2. A number n is called odd if ∃ a ∈ Z such that 2 a + 1 = n . As a first example of a proof, we prove the following: Proposition 3. The sum of two odd numbers is even. Proof. Suppose that x and y are two odd numbers. We may choose a,b ∈ Z such that x = 2 a + 1 and y = 2 b + 1. It follows that x + y = 2 a + 2 b + 2. Since 2 a + 2 b + 2 = 2( a + b + 1), we find that: x + y = 2( a + b + 1) . Hence x + y is even. / Try to follow this template, to prove the following statements: • The sum of an even number and an odd number is odd....
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This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.

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Proofs and Congruence - Math 100 Evens and Odds, First...

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