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Beginning Activity Beginning Activity 2: A Biconditional Statement
A4US

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.
Explain.