Question

Given $p$ : A man is a judge, $q$ : A man is honest.
If $S_1$ : If a man is a judge, then he is honest
$S_2$ : If a man is a judge, then he is not honest
$S_3$ : A man is not a judge or he is honest
$S_4$ : A man is a judge and he is honest
Then

• A. $S_2 \equiv S_3$
• B. $S_1 \equiv S_2$
• C. $S_2 \equiv S_4$
• D. $S_1 \equiv S_3$

Hint:

Memorize the fundamental rule: "If P then Q" ($\text{P} \rightarrow \text{Q}$) can always be rewritten as "Not P or Q" ($\sim\text{P} \vee \text{Q}$). Recognizing this identity allows you to match statements $S_1$ and $S_3$ instantly without building truth tables!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Identify which pair among four compound statements S₁ to S₄ are logically equivalent.

Step 2: Key Formula or Approach:
Convert prose to symbols: "If p then q" = p→q; "not p or q" = ~p∨q; use the identity p→q ≡ ~p∨q.

Step 3: Detailed Explanation:
S₁: p→q; S₂: p→~q; S₃: ~p∨q; S₄: p∧q. Since p→q ≡ ~p∨q, we have S₁ ≡ S₃.

Step 4: Final Answer:
The equivalence is S₁ ≡ S₃, matching option (D).