Quiz 4
Discrete Mathematics · 2 problems · solutions hidden, click to reveal
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Problems
State the definition of an integer being even.
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Answer: An integer \(x\) is even if there is some integer \(a\) such that \(x = 2a\).
Prove that if \(x\) and \(y\) are even integers, then \(3x - 4y\) is even.
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Suppose \(x\) and \(y\) are even. Then there are integers \(a, b\) with \(x = 2a\) and \(y = 2b\). So
\(3x - 4y = 3(2a) - 4(2b) = 6a - 8b = 2(3a - 4b).\)
Since \(3a - 4b\) is an integer, it witnesses that \(3x - 4y\) is even.
\(3x - 4y = 3(2a) - 4(2b) = 6a - 8b = 2(3a - 4b).\)
Since \(3a - 4b\) is an integer, it witnesses that \(3x - 4y\) is even.