Question

Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(x, y \in \mathbb{Z}\). If \(x^{2}(y+3)\) is even, then \(x\) is even or \(y\) is odd.

Short answer

If \(x\) is not even (i.e., odd) and \(y\) is not odd (i.e., even), then \(x^{2}(y+3)\) is not even (i.e., odd). Thus, the given statement 'If \(x^{2}(y+3)\) is even, then \(x\) is even or \(y\) is odd' is true.

Step-by-step solution

Understand the Statement

The given statement is 'If \(x^{2}(y+3)\) is even, then \(x\) is even or \(y\) is odd'. The contrapositive of this statement is 'If \(x\) is not even (i.e., odd) and \(y\) is not odd (i.e., even), then \(x^{2}(y+3)\) is not even (i.e., odd)'. Now, we have to prove the contrapositive statement.

Assume the Contrapositive

We assume that \(x\) is odd and \(y\) is even. The mathematical representation of an odd number is \(2m+1\) and of an even number is \(2n\) where \(m\) and \(n\) are any integers. Therefore, we can take \(x=2m+1\) and \(y=2n\)

Substitute the Values

Now, substitute \(x=2m+1\) and \(y=2n\) in \(x^{2}(y+3)\), we get \((2m+1)^{2}((2n)+3)\ =\ (4m^{2}+4m+1)(2n+3)\), which simplifies to \(8mn^{2}+12mn+4m^{2}+6m+2n+3\), and further simplifies to \(2((4mn^{2}+6mn+2m^{2}+3m+n)+1.5)\)

Conclude the Result

Since \(4mn^{2}+6mn+2m^{2}+3m+n\) is an integer, let's denote it by \(p\). Therefore, the terminus becomes \(2p+1.5\) which cannot be an even number as it doesn't match the form of an even number \(2k\), where \(k\) is an integer. Thus, proving the contrapositive that if \(x\) is not even (i.e., odd) and \(y\) is not odd (i.e., even), then \(x^{2}(y+3)\) is not even (i.e., odd). Hence, the original statement is true.