Consider the following statement. The product of any even integer and any odd integer is even. The hypothesis is --- Select--- The conclusion is --- Select--- (b) The following is a proposed proof of the given statement. 1. Suppose $m$ is any even integer and $n$ is any odd integer. 2. If $m \cdot n$ is even, then by definition of even there exists an integer $r$ such that $m \cdot n=2 r$. 3. Also since $m$ is even, there exists an integer $p$ such that $m=2 p$ by definition of even. 4. And since $n$ is odd, there exists an integer $q$ such that $n=2 q+1$ by definition of odd. 5. Thus, by substitution, $m \cdot n=(2 p)(2 q+1)=2 r$, where $r$ is an integer. 6 . Hence, by definition of even, then, $m \cdot n$ is even, as was to be shown. Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof. Let $S$ be the sentence, "There is an integer $r$ such that $m \cdot n$ equals $2 r . "$ Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction. Thus, the conclusion in Step 5 is not a valid deduction. Step 5 deduces a conclusion that would be true if $S$ were known to be true. The truth of $S$ is not known at the point where Step 5 occurs because the assumption that $m \cdot n$ is even has not been proved. Step 2 states that the truth of $S$ would follow from the assumption that $m \cdot n$ is even. Hence, the proposed proof is circular; it assumes what is to be proved.

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