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An integer n is called even if n = 2m for some integer m, and...

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Advance Math

An integer n is called even if n = 2m for some integer m, and odd if n + 1 is even. Prove the following statements:

(a) An integer cannot be both even and odd.

(b) Every integer is either even or odd.

(c) The sum or product of even integers is an even integer. What can you say about the sum or product of odd integers?

Answer & Explanation Solved by verified expert
4.3 Ratings (748 Votes)
a If an integer is both even and odd there exist integers m and n such that 2m 2n 1 This implies that 1 2mn which means that 1 is even but this is a contradiction since 1 0 1 is odd Hence every integer is either ever or odd b Suppose on the contrary that the set S k Z k is neither even nor odd is    See Answer
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