Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

1.

In Task 4.a from Section 2.2, we constructed a truth table to prove that the biconditional statement, $P\leftrightarrow Q\text{,}$ is logically equivalent to $\left(P\to Q\right)\wedge \left(Q\to P\right)\text{.}$ Complete this exercise if you have not already done so.

2.

Suppose that we want to prove a biconditional statement of the form $P\leftrightarrow Q\text{.}$ Explain a method for completing this proof based on the logical equivalency in Exercise 1.

3.

Let $n$ be an integer. Assume that we have completed the proofs of the following two statements:

• If $n$ is an odd integer, then $n^2$ is an odd integer.

• If $n^2$ is an odd integer, then $n$ is an odd integer.

(See Task 3.c from Section 1.2 and Beginning Activity 1.) Have we completed the proof of the following proposition?

For each integer $n\text{,}$ $n$ is an odd integer if and only if $n^2$ is an odd integer.