Question

If p: it is raining, q: weather is humid, which of the following statements are logically equivalent.
A. If it is not raining then the weather is not humid.
B. It is raining if and only if the weather is humid.
C. It is not true that it is not raining or the weather is humid.
D. It is raining but the weather is not humid.

Answer 1

Hint: This is a problem of mathematical reasoning. We will start by writing each of these statements in their symbolic forms. We will then simplify these symbolic forms. Finally, we will write the truth table for each of these statements and check which statements form the same truth table.

Complete step-by-step answer:
We have been given two statements- p: it is raining and q: weather is humid. We will first try to write these in a symbolic form.
If it is not raining then the weather is not humid = $\text{~}p \to \text{~}q$
It is raining if and only if the weather is humid = $p \leftrightarrow q$
It is not true that it is not raining or the weather is humid = $\text{~}\left( {\text{~}p \vee q} \right)$
It is raining but the weather is not humid = $p \wedge \text{~}q$
We have the symbolic representation of these statements, so we will now try to form the truth table of all these statements. The statements which give the same truth table are logically equivalent, and this will give us our answer.

A. If it is not raining then the weather is not humid.

p	q	~p	~q	∼p ∨ ∼q	~p→∼q
T	T	F	F	T	T
T	F	F	T	F	T
F	T	T	F	T	F
F	F	T	T	T	T

B. It is raining if and only if the weather is humid.

p	q	~p	~q	∼p ∨ ∼q	p↔q
T	T	F	F	T	T
T	F	F	T	F	F
F	T	T	F	T	F
F	F	T	T	T	T

C. It is not true that it is not raining or the weather is humid.

p	q	~p	~q	∼p ∨ ∼q	∼(~p ∨ q)
T	T	F	F	T	F
T	F	F	T	F	T
F	T	T	F	T	F
F	F	T	T	T	F

D. It is raining but the weather is not humid.

p	q	~p	~q	∼p ∨ ∼q	p ∧ ∼q
T	T	F	F	T	F
T	F	F	T	F	T
F	T	T	F	T	F
F	F	T	T	T	F

From these four truth tables, it is clearly visible that the tables formed by options C and D are logically equivalent, and hence the correct options are C and D.

Note: The important thing for students to do here is that they need to memorize the meanings of all the symbols, because they will help in forming the truth tables for each of the statements. For example, in this problem, ~ stands for ‘not’, ∧ stands for ‘and’, ∨ stands for ‘or’ and so on.