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Beginning Activity Beginning Activity 2: Using the Contrapositive
A4US

The following statement was proven in Task 3.c in Section 1.2.

If \(n\) is an odd integer, then \(n^2\) is an odd integer.

Now consider the following proposition:

For each integer \(n\text{,}\) if \(n^2\) is an odd integer, then \(n\) is an odd integer.

1.

After examining several examples, decide whether you think this proposition is true or false.

2.

Try completing the following know-show table for a direct proof of this proposition. The question is, “Can we perform algebraic manipulations to get from the ‘know’ portion of the table to the ‘show’ portion of the table?” Be careful with this! Remember that we are working with integers and we want to make sure that we can end up with an integer \(q\) as stated in Step \(Q1\text{.}\)

Step Know Reason
\(P\) \(n^2\) is an odd integer. Hypothesis
\(P1\) \(\left( \exists k \in \Z \right) \left( n^2 = 2k + 1 \right)\) Definition of “odd integer”
\(\vdots\) \(\vdots\) \(\vdots\)
\(Q1\) \(\left( \exists q \in \Z \right) \left( n=2q \right)\)
\(Q\) \(n\) is an odd integer. Definition of “odd integer”
Step Show Reason

Recall that the contrapositive of the conditional statement \(P \to Q\) is the conditional statement \(\mynot Q \to \mynot P\text{,}\) which is logically equivalent to the original conditional statement. (It might be a good idea to review Beginning Activity 2 from Section 2.2.) Consider the following proposition once again:

For each integer \(n\text{,}\) if \(n^2\) is an odd integer, then \(n\) is an odd integer.

3.

Write the contrapositive of this conditional statement. Please note that “not odd” means “even.” (We have not proved this, but it can be proved using the Division Algorithm in Section 3.5.)

4.

Complete a know-show table for the contrapositive statement from Exercise 3.

5.

By completing the proof in Exercise 4, have you proven the given proposition? That is, have you proven that if \(n^2\) is an odd integer, then \(n\) is an odd integer? Explain.