Question

“If p, then q” is logically equivalent to which of the following?
(I) If q, then p.
(II) If not p, then not q.
(III) If not q, then not p.
(A) None of the above
(B) III only
(C) I and II only
(D) I and III only
(E) I, II and III

Answer 1

Hint:
Firstly, construct the truth tables for the statements “If p, then q”, “If q, then p”, “If not p, then not q” and “If not q, then not p” individually.
Then, after constructing the truth tables of the above statements, compare which of the truth tables are the same as the truth table of “if p, then q”.
Thus, choose the correct answer.

Complete step by step solution:
The given statement is “If p, then q”.
Firstly, we will construct a truth table for the statement “If p, then q” i.e. $p \to q$ .

P	q	$p \to q$
T	T	T
T	F	F
F	T	T
F	F	T

Now, we will draw a truth table for “If q, then p” i.e. $q \to p$ .

p	q	$q \to p$
T	T	T
T	F	T
F	T	F
F	F	T

Also, the truth table for the statement “If not p, then not q” i.e. $ \sim p \to \sim q$ is drawn as

p	q	$ \sim p$	$ \sim q$	$ \sim p \to \sim q$
T	T	F	F	T
T	F	F	T	T
F	T	T	F	F
F	F	T	T	T

Now, the truth table for the statement “If not q, then not p” i.e. $ \sim q \to \sim p$ can be constructed as

p	q	$ \sim p$	$ \sim q$	$ \sim q \to \sim p$
T	T	F	F	T
T	F	F	T	F
F	T	T	F	T
F	F	T	T	T

Now, we have to observe all the above truth tables and see which of the truth tables are the same as the truth table of “if p, then q”.
On observing the truth tables, we come to a conclusion that, the truth table of statement “If not q, then not p” is the same as the truth table of table of statement “If p, then q”.

Thus, option (B) is correct.

Note:
Alternate method:
The statement “If p, then q”, suggests that if p happens, then q will also happen.
The statement if p happens, then q will also happen implies if q won’t happen, then p will not happen too.
Thus, the statement if q won’t happen, then p will not happen too can be written as “If not q, then not p”.
So, option (B) is correct.