Question

Write (a) the contrapositive and (b) the inverse of the following statement.

If $n=17$ then $4n=68$.

Short answer

a. The contrapositive of “If $n=17$ then $4n=68$” is“If $4n\ne 68$then $n\ne 17$”.

b. The inverse of “If $n=17$ then $4n=68$” is“If $n\ne 17$then $4n\ne 68$”.

Step-by-step solution

a.Step 1- Apply the concept of the contrapositive statement.

A contrapositive of any statement is the switching of hypothesis and the conclusion of the inverse of that statement such thatto form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

Step 2- Write the inverse of the statement.

Consider the statementIf $n=17$ then $4n=68$.

Write the inverse of the conditional statement as follows:

If $n\ne 17$ then $4n\ne 68$.

Step 3- Write the contrapositive of the statement.

Interchange the hypothesis and the conclusion of the inverse statement to write the contrapositive of the statement as follows:

If $4n\ne 68$then $n\ne 17$.

Therefore, the contrapositive of the statement is “If $4n\ne 68$then $n\ne 17$”.

b.Step 1- Apply the concept of the inverse of a statement.

To form the inverse of a conditional statement, take the negation of both the hypothesis and the conclusion.

Step 2- Write the inverse of the statement.

Consider the statementIf $n=17$ then $4n=68$.

Take the negation of the hypothesis and the conclusion to write the inverse of the statement.

Step 3- Step description.

The negation of the hypothesis and the conclusion is “If $n\ne 17$ then $4n\ne 68$”.

Therefore, the inverse of the statement is “If $n\ne 17$ then $4n\ne 68$”.