Chapter B.1, Problem 4E

Solution

(a.)

The completed Truth Table.

It has been determined that the completed truth table is as follows:

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

Given:

The incomplete truth table:

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

Concept used:

The truth table shows the truth values of some expression for corresponding truth values of variables.

Calculation:

The given incomplete truth table is:

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

Completing the truth table as follows:

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

Conclusion:

It has been determined that the completed truth table is as follows:

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

(b.)

The completed Truth Table.

It has been determined that the completed truth table is as follows:

$p$	$\sim p$	$p\vee \sim p$	$p\wedge \sim p$
T	F	T	F
F	T	T	F

Given:

The incomplete truth table:

$p$	$\sim p$	$p\vee \sim p$	$p\wedge \sim p$
T
F

Concept used:

The truth table shows the truth values of some expression for corresponding truth values of variables.

Calculation:

The given incomplete truth table is:

$p$	$\sim p$	$p\vee \sim p$	$p\wedge \sim p$
T
F

Completing the truth table as follows:

$p$	$\sim p$	$p\vee \sim p$	$p\wedge \sim p$
T	F	T	F
F	T	T	F

Conclusion:

It has been determined that the completed truth table is as follows:

$p$	$\sim p$	$p\vee \sim p$	$p\wedge \sim p$
T	F	T	F
F	T	T	F

(c.)

If $p$ is logically equivalent to $\sim \left(\sim p\right)$ , based on part (a).

It has been determined that $p$ and $\sim \left(\sim p\right)$ are logically equivalent.

Given:

  $p$ and $\sim \left(\sim p\right)$ .

Concept used:

Two logical expressions are logically equivalent if their truth table are identical.

Calculation:

The completed truth table from part (a) is as follows:

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

It can be seen that the truth table for $p$ and $\sim \left(\sim p\right)$ are identical.

Hence, $p$ and $\sim \left(\sim p\right)$ are logically equivalent.

Conclusion:

It has been determined that $p$ and $\sim \left(\sim p\right)$ are logically equivalent.

(d.)

If $p\vee \sim p$ is logically equivalent to $p\wedge \sim p$ , based on part (b).

It has been determined that $p\vee \sim p$ and $p\wedge \sim p$ are not logically equivalent.

Given:

  $p\vee \sim p$ and $p\wedge \sim p$ .

Concept used:

Two logical expressions are logically equivalent if their truth table are identical.

Calculation:

The completed truth table from part (b) is as follows:

$p$	$\sim p$	$p\vee \sim p$	$p\wedge \sim p$
T	F	T	F
F	T	T	F

It can be seen that the truth table for $p\vee \sim p$ and $p\wedge \sim p$ are not identical.

Hence, $p\vee \sim p$ and $p\wedge \sim p$ are not logically equivalent.

Conclusion:

It has been determined that $p\vee \sim p$ and $p\wedge \sim p$ are not logically equivalent.