## Problem 1056

Let p  and q represent the following simple statements.

p: It is Independence Day.

q: It is July 4th.

Write the following compound statement in its symbolic form.

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

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 4th” 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 4th.

~p                                  ↔              ~q

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

## Problem 1055

Let p and q represent the following simple statements.

p: I jog.

q: I go to the gym.

Write the following compound statement in symbolic form.

I jog or I do not go to the gym.

Solution:-

The given compound statement is disjunction formed by joining two simple statement with the connective “or”.

Examine the first simple statement, “I jog.” This is the same as p in the problem statement.

Thus, “I jog” corresponds to p.

The connective “or” is represented by the symbol  v.

Now examine the second simple statement, “I do not go to the gym.” This is the negation of q in the problem statement . Thus ,” I do not go to the gym” corresponds to ~q.

Therefore , the statement “I jog or I do not go to the gym” corresponds to the symbolic form p v ~ q.