MATH SOLVE

3 months ago

Q:
# Proof by contradiction!Prove that the following statement is true: the sum of two odd numbers is an even number

Accepted Solution

A:

Answer:The proposition: The sum of two odd numbers is an even number is true.Step-by-step explanation:A proof by contradiction is a proof technique that is based on this principle:To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false.Facts that we need:Any even number has the form 2nAny odd number has the form 2n + 1Proposition. The sum of two odd numbers is an even numberProof. Suppose this proposition is false in this case we assume that the sum of two odd numbers is not even. (That would mean that there are two odd numbers out there in the world somewhere that'll give us an odd number when we add them.)Let a, b be odd numbers. Then there exist numbers m, n, such that a = 2m + 1, b = 2n + 1 .Thus a + b = (2m + 1) + (2n + 1) = 2(m + n + 1) which is even. This contradicts the assumption that the sum of two odd numbers is not even.