Question

If the truth value of the expression $[(p \vee q) \wedge (q \to r) \wedge (\sim r)] \to (p \wedge q)$ is False, then truth values of p, q, r are respectively ______.

• A. T, T, T
• B. T, F, F
• C. F, F, F
• D. F, T, T

Hint:

An implication arrow ($\to$) evaluating to False is the greatest gift in a logic problem! It instantly locks the left side to True and the right side to False, collapsing the puzzle immediately.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a complex logical implication statement that evaluates to "False". We must reverse-engineer the required individual truth values (True or False) of propositions $p, q$, and $r$.

Step 2: Detailed Explanation:
The entire logical statement is structured as a massive implication: $X \to Y$.
Where:
Hypothesis $X = [(p \vee q) \wedge (q \to r) \wedge (\sim r)]$
Conclusion $Y = (p \wedge q)$
By the strict rules of mathematical logic, an implication $X \to Y$ is False in one, and only one, specific scenario:
The Hypothesis ($X$) must be True (T), AND the Conclusion ($Y$) must be False (F).
Let's break down the Hypothesis ($X$), which we now know MUST be True:
$X = [(p \vee q) \wedge (q \to r) \wedge (\sim r)] \equiv \text{T}$
For an "AND" ($\wedge$) chain to be True, absolutely every single component must be True.
1. Third component: $(\sim r) \equiv \text{T}$
This immediately dictates that $r$ must be False (F).
2. Second component: $(q \to r) \equiv \text{T}$
We know $r = \text{F}$. The implication $(q \to \text{F})$ can only be True if $q$ is also False. (If $q$ were True, $\text{T} \to \text{F}$ would be False).
Therefore, $q$ must be False (F).
3. First component: $(p \vee q) \equiv \text{T}$
We know $q = \text{F}$. So, $(p \vee \text{F}) \equiv \text{T}$.
This can only happen if $p$ provides the True value.
Therefore, $p$ must be True (T).
Let's verify this against the required Conclusion ($Y$) being False:
$Y = (p \wedge q) = (\text{T} \wedge \text{F}) \equiv \text{F}$.
This perfectly matches our required condition.
The necessary truth values are $p=\text{T}, q=\text{F}, r=\text{F}$.

Step 3: Final Answer:
The truth values are T, F, F, matching option (b).