Question

If p, q, r denote arbitrary statements, then the logically equivalent of the statement \[p \Rightarrow \left( {q \vee r} \right)\] , is
A. \[\left( {p \vee q} \right) \Rightarrow r\]
B. \[\left( {p \Rightarrow q} \right) \vee \left( {p \Rightarrow r} \right)\]
C. \[\left( {p \Rightarrow \sim q} \right) \wedge \left( {p \Rightarrow r} \right)\]
D. \[\left( {p \Rightarrow q} \right) \wedge \left( {p \Rightarrow \sim r} \right)\]

Answer 1

Hint: In this type of question, we have to check each option and simplify it.
A statement with if p, then q is written as p implies q, denoted by $p \Rightarrow q$.
The conditional statement: if p, then q is logically equivalent to negation p or q, denoted by $\left( { \sim p \vee q} \right)$.
The truth table for if p, then q has to be memorized, given as follows:

p	q	$p \Rightarrow q$
0	1	1
0	0	1
1	1	1
1	0	0

The truth table for $\left( { \sim p \vee q} \right)$

p	q	$p \Rightarrow q$	$\left( { \sim p} \right)$	$\left( { \sim p \vee q} \right)$
0	1	1	1	1
0	0	1	1	1
1	1	1	0	1
1	0	0	0	0

 Thus, $p \Rightarrow q = \left( { \sim p \vee q} \right)$

Complete step-by-step answer:
Find the logically equivalent statement of the given statement.
Given statement: \[p \Rightarrow \left( {q \vee r} \right)\]
We know, for: $p \Rightarrow q$, its logically equivalent statement: $\left( { \sim p \vee q} \right)$.
Hence, for: \[p \Rightarrow \left( {q \vee r} \right)\] logically equivalent statement: \[ \sim p \vee \left( {q \vee r} \right)\].
Also, $\left( { \sim p \vee q \vee r} \right)$

Simplify option (A).
\[\left( {p \vee q} \right) \Rightarrow r\]
Its logically equivalent statement: \[ \sim \left( {p \vee q} \right) \vee r\]
Also, \[\left( {\left( { \sim p \wedge \sim q} \right) \vee r} \right)\]; this is not equal to the logically equivalent statement in step (1).

Simplify option (B).
\[\left( {p \Rightarrow q} \right) \vee \left( {p \Rightarrow r} \right)\]
Its logically equivalent statement: \[\left( { \sim p \vee q} \right) \vee \left( { \sim p \vee r} \right)\]
Also, \[\left( { \sim p \vee q \vee \sim p \vee r} \right)\]
We know,
 $
  a \vee a = a \\
  \because \sim p \vee \sim p = \sim p \\
 $
This implies, the logically equivalent statement becomes: \[\left( { \sim p \vee q \vee r} \right)\] ; which is equal to the logically equivalent statement in step (1)

Simplify option (C).
\[\left( {p \Rightarrow \sim q} \right) \wedge \left( {p \Rightarrow r} \right)\]
Its logically equivalent statement: \[\left( { \sim p \vee \sim q} \right) \wedge \left( { \sim p \vee r} \right)\]
We know, $a \vee \left( {b \wedge c} \right) = \left( {a \vee b} \right) \wedge \left( {a \vee c} \right)$ i.e. “or” operation is distributed over “and” operation.
Hence, the logically equivalent statement becomes: \[\left( { \sim p \vee \left( { \sim q \wedge r} \right)} \right)\] ; which is not equal to the logically equivalent statement in step (1).

Simplify option (D).
\[\left( {p \Rightarrow q} \right) \wedge \left( {p \Rightarrow \sim r} \right)\]
Its logically equivalent statement: \[\left( { \sim p \vee q} \right) \wedge \left( { \sim p \vee \sim r} \right)\]
We know, $a \vee \left( {b \wedge c} \right) = \left( {a \vee b} \right) \wedge \left( {a \vee c} \right)$ i.e. “or” operation is distributed over “and” operation.
Hence, the logically equivalent statement becomes: \[\left( { \sim p \vee \left( {q \wedge \sim r} \right)} \right)\] ; which is not equal to the logically equivalent statement in step (1).

So, the correct answer is “Option B”.

Note: The “and” operation is can also be distributed over “or” operation i.e. $a \wedge \left( {b \vee c} \right) = \left( {a \wedge b} \right) \vee \left( {a \wedge c} \right)$
Alternate method: We can solve by using the truth table as well.

Draw the truth table for all the conditional statements given in the question as well as options.
Compare whose values are equal to the given statement is equal to the given option.
Here, for reference, the truth table for the only correct option is given.

p	q	r	$\left( {q \vee r} \right)$	$p \Rightarrow \left( {q \vee r} \right)$	\[p \Rightarrow q\]	\[p \Rightarrow r\]	\[\left( {p \Rightarrow q} \right) \vee \left( {p \Rightarrow r} \right)\]
0	0	0	0	1	1	1	1
0	0	1	1	1	1	1	1
0	1	0	1	1	1	1	1
0	1	1	1	1	1	1	1
1	0	0	0	0	0	0	0
1	0	1	1	1	0	1	1
1	1	0	1	1	1	0	1
1	1	1	1	1	1	1	1

We see that values of \[p \Rightarrow \left( {q \vee r} \right)\] and \[\left( {p \Rightarrow q} \right) \vee \left( {p \Rightarrow r} \right)\] are equal. Thus, \[\left( {p \Rightarrow q} \right) \vee \left( {p \Rightarrow r} \right)\] is the logically equivalent statement of \[p \Rightarrow \left( {q \vee r} \right)\].