Question

The Boolean expression \[\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)\] is equivalent to
A.\[ \sim p \vee q\]
B.\[ \sim q \vee p\]
C.\[p \vee q\]
D.\[\left( { \sim p} \right) \vee ( \sim q)\]

Answer 1

Hint: We will find the equivalent statement for the given Boolean expression with the help of truth table. The truth table that matches with that of the conditions given will be the correct option.

Complete step-by-step answer:
Let’s tabulate the expression.

p	q	\[p \wedge q\]	\[ \sim q\]	\[\left( { \sim q} \right) \vee p\]	\[\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)\]
0	0	0	1	1	1
0	1	0	0	0	0
1	0	0	1	1	1
1	1	1	0	1	1

Now if we observe the table above we will conclude that output is the same as \[\left( { \sim q} \right) \vee p\].
This is one of the options also.
So the correct option is B.
Note: In AND operation output is 1 only if both the inputs are 1.
In OR operation output is 1 if any of the one inputs is 1.
We also can check for other options if we want

p	q	\[p \wedge q\]	\[\left( { \sim q} \right) \vee p\]	\[\left( {p \wedge q} \right) \vee \left( {\left( { \sim q} \right) \vee p} \right)\]	\[ \sim p \vee q\]	\[p \vee q\]	\[ \sim p \vee \sim q\]
0	0	0	1	1	1	0	1
0	1	0	0	0	1	1	1
1	0	0	1	1	0	1	1
1	1	1	1	1	1	1	0

So from the above table we observe that no other option matches with the given Boolean expression output.