Chapter B.1, Problem 10E

Solution

(a.)

Whether the given pair of statements are logically equivalent.

It has been determined that the given pair of statements are not logically equivalent.

Given:

  $\sim \left(p\vee q\right)$ and $\sim p\vee \sim q$

Concept used:

Logical statements are logically equivalent if their truth tables are identical.

Calculation:

The truth table for the given pair of statements is given as:

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

It can be seen that the truth tables for $\sim \left(p\vee q\right)$ and $\sim p\vee \sim q$ are not identical.

Hence, they are not logically equivalent.

Conclusion:

It has been determined that the given pair of statements are not logically equivalent.

(b.)

Whether the given pair of statements are logically equivalent.

It has been determined that the given pair of statements are logically equivalent.

Given:

  $\sim \left(p\vee q\right)$ and $\sim p\wedge \sim q$

Concept used:

Logical statements are logically equivalent if their truth tables are identical.

Calculation:

The truth table for the given pair of statements is given as:

$p$	$q$	$p\vee q$	$\sim \left(p\vee q\right)$	$\sim p$	$\sim q$	$\sim p\wedge \sim q$
F	F	F	T	T	T	T
F	T	T	F	T	F	F
T	F	T	F	F	T	F
T	T	T	F	F	F	F

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

Hence, they are logically equivalent.

Conclusion:

It has been determined that the given pair of statements are logically equivalent.

(c.)

Whether the given pair of statements are logically equivalent.

It has been determined that the given pair of statements are not logically equivalent.

Given:

  $\sim \left(p\wedge q\right)$ and $\sim p\wedge \sim q$

Concept used:

Logical statements are logically equivalent if their truth tables are identical.

Calculation:

The truth table for the given pair of statements is given as:

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

It can be seen that the truth tables for $\sim \left(p\wedge q\right)$ and $\sim p\wedge \sim q$ are not identical.

Hence, they are not logically equivalent.

Conclusion:

It has been determined that the given pair of statements are not logically equivalent.

(d.)

Whether the given pair of statements are logically equivalent.

It has been determined that the given pair of statements are logically equivalent.

Given:

  $\sim \left(p\wedge q\right)$ and $\sim p\vee \sim q$

Concept used:

Logical statements are logically equivalent if their truth tables are identical.

Calculation:

The truth table for the given pair of statements is given as:

$p$	$q$	$p\wedge q$	$\sim \left(p\wedge q\right)$	$\sim p$	$\sim q$	$\sim p\vee \sim q$
F	F	F	T	T	T	T
F	T	F	T	T	F	T
T	F	F	T	F	T	T
T	T	T	F	F	F	F

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

Hence, they are logically equivalent.

Conclusion:

It has been determined that the given pair of statements are logically equivalent.