Question

Consider the following logical statements:
R: If \(p \rightarrow q\) is false, then \(p \vee q\) is false.
S: If \(p \leftrightarrow q\) is false, then \(p \vee q\) is false.
Evaluate the truth values of R and S.

• A. Both R and S are true
• B. R is true and S is false
• C. R is false and S is true
• D. Both R and S are false

Hint:

An implication \( P \rightarrow Q \) is only false when the premise \( P \) is True and the conclusion \( Q \) is False.
If you can find one case where the premise is met but the conclusion is not, the statement is false.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to determine if the given implications are tautologies or false statements by analyzing the truth conditions of the premises.

Step 2: Key Formula or Approach:
1. \( p \rightarrow q \) is false only when \( p = T \) and \( q = F \).
2. \( p \leftrightarrow q \) is false when \( p \) and \( q \) have different truth values (\( T,F \) or \( F,T \)).
3. \( p \vee q \) is false only when both \( p \) and \( q \) are false.

Step 3: Detailed Explanation:
Evaluating R:
Premise: \( p \rightarrow q \) is false. This implies \( p = T \) and \( q = F \).
Conclusion: \( p \vee q \). If \( p = T \) and \( q = F \), then \( p \vee q = T \vee F = T \).
Since the conclusion is True when the premise is True, the statement "then \( p \vee q \) is false" is False. Thus, R is False.

Evaluating S:
Premise: \( p \leftrightarrow q \) is false. This occurs if (\( p=T, q=F \)) or (\( p=F, q=T \)).
Case 1: \( p=T, q=F \). Then \( p \vee q = T \vee F = T \).
Case 2: \( p=F, q=T \). Then \( p \vee q = F \vee T = T \).
In both cases where the premise is true, the conclusion \( p \vee q \) is True. Thus, the statement "then \( p \vee q \) is false" is False. Thus, S is False.

Step 4: Final Answer:
Both R and S are false statements.