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      <text>3.3.2 (Example of contrapositive proof)</text>
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      <text><![CDATA[<h3>A contrapositive proof</h3>


<p><b>Lemma:</b> If \(a^2\) is an even number then \(a\) is even.
<div class="proof">
<div class="proof-line">
 <div class="proof-num">1.</div>
 <div class="proof-step">Let \(A\) be the statement "\(a^2\) is even" and \(B\) be the statement "\(a\) is even".</div>
</div>
<div class="proof-line">
 <div class="proof-num">2.</div>
 <div class="proof-step">We will prove the contrapositive of the statement, that is we will prove the statement (directly):
<p>"If \(a\) is odd then \(a^2\) is odd"</p></div>
</div>
<div class="proof-line">
 <div class="proof-num">3.</div>
 <div class="proof-step">Suppose that \(a\) is odd then we can write \(a = 2k+1\), where \(k\) is some integer.</div>
</div>
<div class="proof-line">
 <div class="proof-num">4.</div>
 <div class="proof-step">It follows that:
\[
\begin{align*}
a^2 &= (2k+1)(2k+1)
\\
&= (4k^2 + 2k+2k) + 1
\\
&= 2(2k^2 + 2k) + 1.
\end{align*}
\]
</div>
</div>
<div class="proof-line">
 <div class="proof-num">5.</div>
 <div class="proof-step">Since, \(k\) is an integer, so is \((2k^2+2k)\) is an integer and thus \(a^2\) can be written in the form \(2(\mbox{integer})+1\).</div>
</div>
<div class="proof-line">
 <div class="proof-num">6.</div>
 <div class="proof-step">Hence \(a^2\) is odd and since we have proven the contrapositive of the given lemma, the proof is complete.</div>
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