Question

a) Describe a way to prove the biconditional\(p \to q\).

b) Prove the statement: “The integer\(3n + 2\)is odd if and only if the integer\(9n + 5\)is even, where n is an integer.”

Short answer

a) The biconditional\(p \to q\) .

b) “The integral value\(3n + 2\)is odd if and only if the integral value \(9n + 5\)is even, where n is an integral value.”

Step-by-step solution

Introduction

A relation between two statements or propositions which is true only when both the propositions are simultaneously true and is false when both are simultaneously false.

(a) Definition of Biconditional Statement

“Assume that p and q are propositions. The biconditional statement\(p \to q\)is the proposition “p if and only if q”.

The biconditional statement\(p \leftrightarrow q\) is the proposition “p if and only if q”.

The biconditional statement \(p \leftrightarrow q\)is true when p and q have the same truth value and is false otherwise.

So, to prove the biconditional statement\(p \leftrightarrow q\) “prove that both the conditional statements\(p \to q\)and\(q \to p \)are true”.

If any of the two conditional statements are false, the biconditional statement\(p \leftrightarrow q\)will be false.

(b) Proof of the given statement

Consider the following Biconditional Statement,

“The integral value\(3n + 2\)is odd if and only if the integral value\(9n + 5\)is even, where n is an even integral value.”

Assume that p and q represent the following propositions:

\(p\): The integer\(3n + 2\)is odd.

\(q\): The integer\(9n + 5\)is even.

Prove that the conditional statement\(p \to q\)is true. That is, if p is true then q is also true.

Assume that the proposition p is true.

So, the integral value\(3n + 2\)is odd.

\( \Rightarrow \)\(3n\)will be odd.

\( \Rightarrow \) The integral value\(n\)will be odd.

\( \Rightarrow \)The integral value\(9n\)will be odd.

\( \Rightarrow \)The integral value\(9n + 4\)will be odd.

\( \Rightarrow \)The integral value\(9n + 5\)will be odd.

\( \Rightarrow \)The proposition q is true.

Thus, the conditional statement\(p \to q\)is true.

Prove that the conditional statement\(q \to p \)is true. That is, if q is rue then p is also true.

Assume that the propositionq is true.

So, the integral value\(9n + 5\)is even.

\( \Rightarrow \)The integral value\(9n\)will be odd.

\( \Rightarrow \) The integral value\(n\)will be odd.

\( \Rightarrow \)The integral value\(3n\)will be odd.

\( \Rightarrow \)The integral value\(3n + 2\)will be odd.

\( \Rightarrow \)The proposition p is true.

Thus, the conditional statement\(q \to p \)is true.

As both the conditional statements\(p \to q\)and\(q \to p \)are true, the biconditional statement\(p \leftrightarrow q\)is proved.

Therefore, “The integral value\(3n + 2\)is odd if and only if the integral value \(9n + 5\)is even, where n is an integral value.”