Give a proof by contradiction of the following: “If x and y are even integers, then xy is even”.  DISCRETE STUCTURES

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Give a proof by contradiction of the following: “If x and y are even integers, then
xy is even”. 

DISCRETE STUCTURES 

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Step 1: Step 1:

To prove the statement "If x and y are even integers, then xy is even" by contradiction, we will assume the opposite and show that it leads to a contradiction. That is, we will assume that x and y are even integers, but xy is not even (i.e., xy is odd).

Assume:

  1. x and y are even integers.
  2. xy is not even (i.e., xy is odd).

Since x and y are even integers, we can express them as:

x space equals space 2 a, where a is an integer.

y space equals space 2 b, where b is an integer.

Now, let's consider the product xy:

x y space equals space left parenthesis 2 a right parenthesis left parenthesis 2 b right parenthesis space equals space 4 a b.

Since xy is not even (it's odd), we can express it as an odd integer plus 1:

x y space equals space 4 a b space plus space 1.

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