Chapter B.2, Problem 10E

Solution

(a.)

To Show: The given expression is a tautology, using truth table.

It has been determined that the given expression is a tautology.

Given:

  $\left(p\to q\right)\to \left[\left(p\wedge r\right)\to q\right]$

Concept used:

If the truth value of an expression in its truth table is always true, then it is a tautology.

Calculation:

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

The truth table for the given expression is as follows:

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

It can be seen that the truth value for the given expression is always true.

Hence, the given expression is a tautology.

Conclusion:

It has been determined that the given expression is a tautology.

(b.)

To Show: The given expression is a tautology, using truth table.

It has been determined that the given expression is a tautology.

Given:

  $\left[\left(p\to q\right)\wedge p\right]\to q$

Concept used:

If the truth value of an expression in its truth table is always true, then it is a tautology.

Calculation:

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

The truth table for the given expression is as follows:

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

It can be seen that the truth value for the given expression is always true.

Hence, the given expression is a tautology.

Conclusion:

It has been determined that the given expression is a tautology.

(c.)

To Show: The given expression is a tautology, using truth table.

It has been determined that the given expression is a tautology.

Given:

  $\left[\left(p\to q\right)\wedge \sim q\right]\to \sim q$

Concept used:

If the truth value of an expression in its truth table is always true, then it is a tautology.

Calculation:

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

The truth table for the given expression is as follows:

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

It can be seen that the truth value for the given expression is always true.

Hence, the given expression is a tautology.

Conclusion:

It has been determined that the given expression is a tautology.

(d.)

To Show: The given expression is a tautology, using truth table.

It has been determined that the given expression is a tautology.

Given:

  $\left[\left(p\to q\right)\wedge \left(q\to r\right)\right]\to \left(p\to r\right)$

Concept used:

If the truth value of an expression in its truth table is always true, then it is a tautology.

Calculation:

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

The truth table for the given expression is as follows:

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

It can be seen that the truth value for the given expression is always true.

Hence, the given expression is a tautology.

Conclusion:

It has been determined that the given expression is a tautology.