Chapter B.2, Problem 3E

Solution

(a.)

The converse, inverse and contrapositive of the given implication.

It has been determined that the converse of the given statement is “If you are good in sports, then you eat Meaties”; the inverse of the given statement is “If you do not eat Meaties, then you are not good in sports”; and the contrapositive of the given statement is “If you are not good in sports, then you do not eat Meaties.”

Given:

If you eat Meaties, then you are good in sports.

Concept used:

The 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 $p$ ”.

Calculation:

The given statement is “If you eat Meaties, then you are good in sports.”

The converse of the given statement is “If you are good in sports, then you eat Meaties.”

The inverse of the given statement is “If you do not eat Meaties, then you are not good in sports.”

The contrapositive of the given statement is “If you are not good in sports, then you do not eat Meaties.”

Conclusion:

It has been determined that the converse of the given statement is “If you are good in sports, then you eat Meaties”; the inverse of the given statement is “If you do not eat Meaties, then you are not good in sports”; and the contrapositive of the given statement is “If you are not good in sports, then you do not eat Meaties.”

(b.)

The converse, inverse and contrapositive of the given implication.

It has been determined that the converse of the given statement is “If you do not like mathematics, then you do not like this book”; the inverse of the given statement is “If you like this book, then you like mathematics”; and the contrapositive of the given statement is “If you like mathematics, then you like this book.”

Given:

If you do not like this book, then you do not like mathematics.

Concept used:

The 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 $p$ ”.

Calculation:

The given statement is “If you do not like this book, then you do not like mathematics.”

The converse of the given statement is “If you do not like mathematics, then you do not like this book.”

The inverse of the given statement is “If you like this book, then you like mathematics.”

The contrapositive of the given statement is “If you like mathematics, then you like this book.”

Conclusion:

It has been determined that the converse of the given statement is “If you do not like mathematics, then you do not like this book”; the inverse of the given statement is “If you like this book, then you like mathematics”; and the contrapositive of the given statement is “If you like mathematics, then you like this book.”

(c.)

The converse, inverse and contrapositive of the given implication.

It has been determined that the converse of the given statement is “If you have cavities, then you do not use Ultra Brush toothpaste”; the inverse of the given statement is “If you use Ultra Brush toothpaste, then you do not have cavities”; and the contrapositive of the given statement is “If you do not have cavities, then you use Ultra Brush toothpaste.”

Given:

If you do not use Ultra Brush toothpaste, then you have cavities.

Concept used:

The 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 $p$ ”.

Calculation:

The given statement is “If you do not use Ultra Brush toothpaste, then you have cavities.”

The converse of the given statement is “If you have cavities, then you do not use Ultra Brush toothpaste.”

The inverse of the given statement is “If you use Ultra Brush toothpaste, then you do not have cavities.”

The contrapositive of the given statement is “If you do not have cavities, then you use Ultra Brush toothpaste.”

Conclusion:

It has been determined that the converse of the given statement is “If you have cavities, then you do not use Ultra Brush toothpaste”; the inverse of the given statement is “If you use Ultra Brush toothpaste, then you do not have cavities”; and the contrapositive of the given statement is “If you do not have cavities, then you use Ultra Brush toothpaste.”

(d.)

The converse, inverse and contrapositive of the given implication.

It has been determined that the converse of the given statement is “If your grades are high, then you are good at logic”; the inverse of the given statement is “If you are not good at logic, then your grades are not high”; and the contrapositive of the given statement is “If your grades are not high, then you are not good at logic.”

Given:

If you are good at logic, then your grades are high.

Concept used:

The 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 $p$ ”.

Calculation:

The given statement is “If you are good at logic, then your grades are high.”

The converse of the given statement is “If your grades are high, then you are good at logic.”

The inverse of the given statement is “If you are not good at logic, then your grades are not high.”

The contrapositive of the given statement is “If your grades are not high, then you are not good at logic.”

Conclusion:

It has been determined that the converse of the given statement is “If your grades are high, then you are good at logic”; the inverse of the given statement is “If you are not good at logic, then your grades are not high”; and the contrapositive of the given statement is “If your grades are not high, then you are not good at logic.”