Chapter 2, Problem 1NT

NOW TRY THIS

a. Use truth tables to prove Theorem $2-1(b)$, the second De Morgan’s Law for logic.

b. Write a sentence showing the use of De Morgan’s second law in everyday language.

c.

To determine

(a)

To prove:

The second De Morgan’s Law for Logic using truth tables.

Answer to Problem 1NT

Solution:

So, the required answer is

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

Explanation of Solution

Given:

The second De Morgan’s Law for Logic

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

Approach:

The second De Morgan’s Law for Logic

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

Calculation:

Hence, the truth table is

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

To determine

(b)

To explain:

The use of De Morgan’s Second Law in everyday language.

Answer to Problem 1NT

Solution:

Let p be “I go the school” and q be “I write a letter”. De Morgan’s Second law of logic might be interpreted as “It is not the case that I go to the school or I write a letter,” has the same meaning as “I do not go the school and I do not write a letter.”

Explanation of Solution

Given:

The second De Morgan’s Law for Logic

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

Approach:

Let p be “I go the school” and q be “I write a letter”. De Morgan’s Second law of logic might be interpreted as “It is not the case that I go to the school or I write a letter” has the same meaning as “I do not go the school and I do not write a letter.”