Chapter B.2, Problem 9E

Solution

A statement logically equivalent to the given statement.

It has been determined that a statement logically equivalent to the given statement is “If a number is not a multiple of $4$ , then it is not a multiple of $8$ .”

Given:

If a number is a multiple of $8$ , then it is a multiple of $4$ .

Concept used:

The conditional statement “If $p$ , then $q$ ” has converse of the form “If $q$ , then $p$ ”, inverse of the form “If not $p$ , then not $q$ ” and contrapositive of the form “If not $q$ , then not $p$ ”.

An implication and its contrapositive are logically equivalent. The converse and the inverse of a conditional statement are logically equivalent.

Calculation:

The given statement is:

“If a number is a multiple of $8$ , then it is a multiple of $4$ .”

As discussed, an implication and its contrapositive are logically equivalent.

The contrapositive of the given statement is:

“If a number is not a multiple of $4$ , then it is not a multiple of $8$ .”

Then, the above statement is logically equivalent to the given statement.

Conclusion:

It has been determined that a statement logically equivalent to the given statement is “If a number is not a multiple of $4$ , then it is not a multiple of $8$ .”