Chapter B.2, Problem 2E

Solution

(a.)

The truth table for the given expression.

The truth table for the given expression has been constructed.

Given:

  $p\to \left(p\vee q\right)$

Concept used:

The truth table of an expression can be obtained by considering all possible truth values of the variables in the expression.

Calculation:

The given expression is $p\to \left(p\vee q\right)$ .

The truth table for this expression is as follows:

$p$	$q$	$p\vee q$	$p\to \left(p\vee q\right)$
T	T	T	T
T	F	T	T
F	T	T	T
F	F	F	T

This is the required truth table.

Conclusion:

The truth table for the given expression has been constructed.

(b.)

The truth table for the given expression.

The truth table for the given expression has been constructed.

Given:

  $\left(p\wedge q\right)\to q$

Concept used:

The truth table of an expression can be obtained by considering all possible truth values of the variables in the expression.

Calculation:

The given expression is $\left(p\wedge q\right)\to q$ .

The truth table for this expression is as follows:

$p$	$q$	$p\wedge q$	$\left(p\wedge q\right)\to q$
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

This is the required truth table.

Conclusion:

The truth table for the given expression has been constructed.

(c.)

The truth table for the given expression.

The truth table for the given expression has been constructed.

Given:

  $p\leftrightarrow \sim \left(\sim p\right)$

Concept used:

The truth table of an expression can be obtained by considering all possible truth values of the variables in the expression.

Calculation:

The given expression is $p\leftrightarrow \sim \left(\sim p\right)$ .

The truth table for this expression is as follows:

$p$	$\sim p$	$\sim \left(\sim p\right)$	$p\leftrightarrow \sim \left(\sim p\right)$
T	F	T	T
F	T	F	T

This is the required truth table.

Conclusion:

The truth table for the given expression has been constructed.

(d.)

The truth table for the given expression.

The truth table for the given expression has been constructed.

Given:

  $\sim \left(p\to q\right)$

Concept used:

The truth table of an expression can be obtained by considering all possible truth values of the variables in the expression.

Calculation:

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

The truth table for this expression is as follows:

$p$	$q$	$p\to q$	$\sim \left(p\to q\right)$
T	T	T	F
T	F	F	T
F	T	T	F
F	F	T	F

This is the required truth table.

Conclusion:

The truth table for the given expression has been constructed.