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 Concept:
The problem requires evaluating the truth conditions of basic logical operators such as implication (\( \rightarrow \)), biconditional (\( \leftrightarrow \)), and disjunction (\( \lor \)).
A logical statement claiming an outcome based on a condition is false if the specified condition holds but the outcome does not.
Step 2: Key Formula or Approach:
- Implication (\( p \rightarrow q \)) is false only when the hypothesis \( p \) is True (T) and the conclusion \( q \) is False (F).
- Biconditional (\( p \leftrightarrow q \)) is false when \( p \) and \( q \) have opposite truth values (i.e., one is True and the other is False).
- Disjunction (\( p \lor q \)) is false only when both \( p \) and \( q \) are False.
Step 3: Detailed Explanation:
Let's analyze statement R:
"If \( p \rightarrow q \) is false, then \( p \lor q \) is false."
The premise states that \( p \rightarrow q \) is false.
This occurs exclusively when \( p = \text{True (T)} \) and \( q = \text{False (F)} \).
Given \( p = \text{T} \) and \( q = \text{F} \), we evaluate \( p \lor q \):
\[ \text{T} \lor \text{F} \equiv \text{True (T)} \]
However, statement R claims that \( p \lor q \) is false.
Since the actual result is True, statement R is false.
Now, let's analyze statement S:
"If \( p \leftrightarrow q \) is false, then \( p \lor q \) is false."
The premise states that \( p \leftrightarrow q \) is false.
This happens in exactly two cases:
Case 1: \( p = \text{True (T)} \) and \( q = \text{False (F)} \)
Case 2: \( p = \text{False (F)} \) and \( q = \text{True (T)} \)
Let's evaluate \( p \lor q \) for both cases:
For Case 1: \( \text{T} \lor \text{F} \equiv \text{True (T)} \)
For Case 2: \( \text{F} \lor \text{T} \equiv \text{True (T)} \)
In both possible scenarios, \( p \lor q \) evaluates to True.
However, statement S claims that \( p \lor q \) is false.
Since the actual result is always True, statement S is false.
Step 4: Final Answer:
Both logical statements R and S are false, which corresponds to option (D).