Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

1.

The proposition is a conditional statement. What is the hypothesis of this conditional statement? What is the conclusion of this conditional statement?

2.

If $x=2$ and $y=3\text{,}$ then $x\cdot y=6\text{,}$ and 6 is an even integer. Does this example prove that the proposition is false? Explain.

3.

If $x=5$ and $y=3\text{,}$ then $x\cdot y=15\text{.}$ Does this example prove that the proposition is true? Explain.

4.

To start a proof of this proposition, we will assume that the hypothesis of the conditional statement is true. So in this case, we assume that both $x$ and $y$ are odd integers. We can then use the definition of an odd integer to conclude that there exists and integer $m$ such that $x=2m+1\text{.}$ Now use the definition of an odd integer to make a conclusion about the integer $y\text{.}$

Note: The definition of an odd integer says that a certain other integer exists. This definition may be applied to both $x$ and $y\text{.}$ However, do not use the same letter in both cases. To do so would imply that $x=y$ and we have not made that assumption. To be more specific, if $x=2m+1$ and $y=2m+1\text{,}$ then $x=y\text{.}$

5.

We need to prove that if the hypothesis is true, then the conclusion is true. So, in this case, we need to prove that $x\cdot y$ is an odd integer. At this point, we usually ask ourselves a so-called backward question. In this case, we ask, “Under what conditions can we conclude that $x\cdot y$ is an odd integer?” Use the definition of an odd integer to answer this question.