## Problem 1056

Let p and q represent the following simple statements.

p: It is Independence Day.

q: It is July 4^{th}.

Write the following compound statement in its symbolic form.

It is not Independence Day if and only if it not July 4^{th}.

**Solution:-**

The given compound statement is a biconditional statement.

A biconditional statement is a compound statement formed by joining two simple statements with the connective if and only if.

“It is not Independence Day” is the first simple statement in the biconditional; it precedes the connective if and only if.

Since this statement is the negation of the statement represented by p, its symbolic form is ~p.

“It is not July 4^{th}” is the second simple statement in the biconditional; it follows the connective if and only if.

Since this statement is the negation of the statement represented by q, its symbolic form is ~q.

The symbol that represents the connective if and only if is ↔.

To write the conditional statement in symbolic form, begin by replacing the first simple statement with its symbolic form. Then, replace the connective if and only if by the symbol that represents it. Finally, write the second statement in symbolic form.

It is not Independence Day if and only if it is not July 4^{th}.

~p ↔ ~q

The symbolic form for the given biconditional statement is ~p ↔~q.