Consider the following theorems and their proofs. Unfortunately, each proof has
ID: 3602653 • Letter: C
Question
Consider the following theorems and their proofs. Unfortunately, each proof has a mistake in it. Find the mistake in each proof, and explain why it is a mistake (you do not have to write a correct proof). a) Theorem: The sum of two odd integers is even. Proof: Suppose x and y are two odd integers. If x+y is even, then by the definition of even, there exists an integer k such that x y-2k. Since x and y are odd, there exists an integer m such that x2m+1, and an integer n such that y 2n+1. Therefore x + y = (2m+1 ) + (2n+1) = 2k By the definition of "even", x +y is even. QEDExplanation / Answer
Solution:
Correct proof is
Proof:
Let x and y be odd integers. By definition of odd we have that x = 2n + 1 and
y = 2m + 1. Consider the sum x + y = (2n + 1) + (2m +1) = 2n + 2m +2 = 2k, where
k = n + m + 1 is an integer. Therefore by definition of even we have shown that
x + y is even and my hypothesis is true.
In the given proof hypothesis is not proved and the point k = n + m + 1 is also not mentioned.
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