Question

Which of the following statement pattern is a contradiction?

• A. S₄ ≡ (∼p ˄ q) ˅ (∼q)
• B. S₂ ≡ (p →q) ˅ (p ˄ ∼q)
• C. S₁ ≡ (∼p ˅ ∼q) ˅ (p ˅ ∼q)
• D. S₃ ≡ (∼p ˄ q) ˄ (∼q)

Answer 1

The Correct Option is D

Detailed Solution:

What is a Contradiction?
In logic, a contradiction is a compound statement that is always false — no matter what truth values you assign to its variables.

Let’s analyze S₃:

S₃ ≡ (~p ∧ q) ∧ (¬q)

This is a conjunction, which means both parts must be true for the whole statement to be true. Let’s look closely:

• ~p ∧ q is true when p is false and q is true.
• ¬q (i.e., not q) is true only when q is false.

But this leads to a contradiction:

• The first part requires q to be true.
• The second part requires q to be false.

It's impossible for q to be both true and false at the same time. So, there is no truth assignment that makes this entire expression true.

Therefore, S₃ is always false — a contradiction.

Truth Table:

As you can see, the final column (S₃) is always false. Hence, it is a contradiction.