Question

Neither p nor q is equivalent to
(a) \[\sim p\wedge \sim q\]
(b) \[\sim \left( p\wedge q \right)\]
(c) \[\left( \sim p\vee q \right)\wedge \left( p\vee \sim q \right)\]
(d) None

Answer 1

Hint: We can write neither p nor q as all elements that are not included in p and q. We have to draw a truth table in binary form, that is, 0 and 1 form. 1 denotes the presence of an element and 0 is the absence of an element. $\sim p$ will be the opposite of p (i.e., not included in p), that is, if p is 1 $\sim p$ will be 0 and vise-versa. Similarly, we have to write for q. We have to get a 1 in the ‘neither p nor q’ section. When an AND operator is performed, this happens only when $\sim p$ and $\sim q$ gets a 1.

Complete step by step answer:
We have to find neither p nor q. This means that all elements that are not included in p and q. We can denote the elements not included in p (NOT p) as $\sim p$ and the elements not included in q (Not q) as $\sim q$ .
Let us write a logical truth table in binary form, that is, 0 and 1 form. In this example, we will consider that 1 is the presence of an element and 0 is the absence of an element. $\sim p$ will be the opposite of p, that is, if p is 1 $\sim p$ will be 0 and vise-versa. Similarly, for q.

p	$\sim p$	q	$\sim q$
0	1	0	1
1	0	1	0

We have seen that neither p nor q means that all elements that are not included in p and q. ‘AND’ in binary operator is denoted by $\wedge $ . We have to get a 1 in ‘neither p nor q’ section. When an AND operator is performed, this happens only when $\sim p$ and $\sim q$ gets a 1. 1 in $\sim p$ and $\sim q$ implies that we are concentrating on elements that are not in p and not in q. The following table shows all the possible values for p and q.

p	q	$\sim p$	$\sim q$	$\sim p\wedge \sim q$
0	0	1	1	1
0	1	1	0	0
1	0	0	1	0
1	1	0	0	0

Hence, neither p nor q is denoted as $\sim p\wedge \sim q$ .
So, the correct answer is “Option a”.

Note: Students must know the notations used in binary operations. ‘AND’ is denoted as $\wedge $ , that is, consider P and Q. We will denote this as $P\wedge Q$ or $P\cdot Q$ or $P\And Q$ . ‘OR’ in binary operator is denoted as $\vee $ , that is, $P\vee Q$ or $P+Q$ .