Give a proof by contradiction of the following: “If x and y are even integers, then xy is even”. DISCRETE STUCTURES
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
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Give a proof by contradiction of the following: “If x and y are even integers, then
xy is even”.
DISCRETE STUCTURES
Expert Solution
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:
- x and y are even integers.
- xy is not even (i.e., xy is odd).
Since x and y are even integers, we can express them as:
, where a is an integer.
, where b is an integer.
Now, let's consider the product xy:
Since xy is not even (it's odd), we can express it as an odd integer plus 1:
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