Chapter B.2, Problem 8E

Solution

(a.)

Whether the given statement is logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

It has been determined that the given statement is not logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

Given:

If every digit of a number is not $6$ , then the number is not divisible by $3$ .

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 reference statement is given as:

“If every digit of a number is $6$ , then the number is divisible by $3$ .”

The given statement is:

“If every digit of a number is not $6$ , then the number is not divisible by $3$ .”

Clearly the given statement is inverse of the reference statement.

As discussed, an implication and its inverse are not logically equivalent.

Hence, the given statement is not logically equivalent to the reference statement.

Conclusion:

It has been determined that the given statement is not logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

(b.)

Whether the given statement is logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

It has been determined that the given statement is logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

Given:

If a number is not divisible by $3$ , then some digit of the number is not $6$ .

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 reference statement is given as:

“If every digit of a number is $6$ , then the number is divisible by $3$ .”

The given statement is:

“If a number is not divisible by $3$ , then some digit of the number is not $6$ .”

Clearly the given statement is contrapositive of the reference statement.

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

Hence, the given statement is logically equivalent to the reference statement.

Conclusion:

It has been determined that the given statement is logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

(c.)

Whether the given statement is logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

It has been determined that the given statement is not logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”

Given:

If a number is divisible by $3$ , then every digit of the number is $6$ .

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 reference statement is given as:

“If every digit of a number is $6$ , then the number is divisible by $3$ .”

The given statement is:

“If a number is divisible by $3$ , then every digit of the number is $6$ .”

Clearly the given statement is converse of the reference statement.

As discussed, an implication and its converse are not logically equivalent.

Hence, the given statement is not logically equivalent to the reference statement.

Conclusion:

It has been determined that the given statement is not logically equivalent to the statement “If every digit of a number is $6$ , then the number is divisible by $3$ .”