Chapter B.2, Problem 6E

Solution

(a.)

The truth value of the given expression.

It has been determined that $\sim p\to \sim q$ is true.

Given:

  $p$ is false and $q$ is false.

The expression, $\sim p\to \sim q$ .

Concept used:

The truth value of an expression can be determined using the truth value of the variables in the expression.

Calculation:

It is given that $p$ is false and $q$ is false.

Then, it follows that $\sim p$ is true and $\sim q$ is true, which implies that $\sim p\to \sim q$ is true.

Conclusion:

It has been determined that $\sim p\to \sim q$ is true.

(b.)

The truth value of the given expression.

It has been determined that $\sim \left(p\to q\right)$ is false.

Given:

  $p$ is false and $q$ is false.

The expression, $\sim \left(p\to q\right)$ .

Concept used:

The truth value of an expression can be determined using the truth value of the variables in the expression.

Calculation:

It is given that $p$ is false and $q$ is false.

Then, it follows that $p\to q$ is true, which implies that $\sim \left(p\to q\right)$ is false.

Conclusion:

It has been determined that $\sim \left(p\to q\right)$ is false.

(c.)

The truth value of the given expression.

It has been determined that $\left(p\vee q\right)\to \left(p\wedge q\right)$ is true.

Given:

  $p$ is false and $q$ is false.

The expression, $\left(p\vee q\right)\to \left(p\wedge q\right)$ .

Concept used:

The truth value of an expression can be determined using the truth value of the variables in the expression.

Calculation:

It is given that $p$ is false and $q$ is false.

Then, it follows that $p\vee q$ is false and $p\wedge q$ is false, which implies that $\left(p\vee q\right)\to \left(p\wedge q\right)$ is true.

Conclusion:

It has been determined that $\left(p\vee q\right)\to \left(p\wedge q\right)$ is true.

(d.)

The truth value of the given expression.

It has been determined that $p\to \sim p$ is true.

Given:

  $p$ is false and $q$ is false.

The expression, $p\to \sim p$ .

Concept used:

The truth value of an expression can be determined using the truth value of the variables in the expression.

Calculation:

It is given that $p$ is false and $q$ is false.

Then, it follows that $\sim p$ is true, which implies that $p\to \sim p$ is true.

Conclusion:

It has been determined that $p\to \sim p$ is true.

(e.)

The truth value of the given expression.

It has been determined that $\left(p\vee \sim p\right)\to p$ is false.

Given:

  $p$ is false and $q$ is false.

The expression, $\left(p\vee \sim p\right)\to p$ .

Concept used:

The truth value of an expression can be determined using the truth value of the variables in the expression.

Calculation:

It is given that $p$ is false and $q$ is false.

Then, it follows that $\sim p$ is true, which implies that $p\vee \sim p$ is true, which further implies that $\left(p\vee \sim p\right)\to p$ is false.

Conclusion:

It has been determined that $\left(p\vee \sim p\right)\to p$ is false.

(f.)

The truth value of the given expression.

It has been determined that $\left(p\vee q\right)\leftrightarrow \left(p\wedge q\right)$ is true.

Given:

  $p$ is false and $q$ is false.

The expression, $\left(p\vee q\right)\leftrightarrow \left(p\wedge q\right)$ .

Concept used:

The truth value of an expression can be determined using the truth value of the variables in the expression.

Calculation:

It is given that $p$ is false and $q$ is false.

Then, it follows that $p\vee q$ is false and $p\wedge q$ is false, which implies that $\left(p\vee q\right)\leftrightarrow \left(p\wedge q\right)$ is true.

Conclusion:

It has been determined that $\left(p\vee q\right)\leftrightarrow \left(p\wedge q\right)$ is true.