If the truth value of the Boolean expression ((p ∨ q) ∧ (q → r) ∧ (∼ r)) → (p ∧ q) is false then the

Q: Wait, why is $A \rightarrow B$ false only when $A=T$ and $B=F$?

Think of an implication as a promise. If I say, 'If it rains, I will carry an umbrella' ($p \rightarrow q$), the only way I have lied is if it rains (True) but I do not carry an umbrella (False). In all other scenarios—it doesn't rain, or I carry an umbrella anyway—the promise holds. Thus, the expression is only false when the antecedent is True and the consequent is False.

Q: Can I start by assuming $p=T$ and testing options?

You could, but that is trial and error, which is risky in competitive exams. Logic is about deduction. By breaking the expression into its components, you solve it systematically without guessing. It is faster and more reliable.

Q: Is there ever a case where the truth values are not unique?

In this specific problem, the constraints are tight. Because we are given that the entire expression is False, we are forced into a single, unique scenario for the antecedent and consequent. This uniqueness is what makes the problem solvable.

Question

If the truth value of the Boolean expression $((p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)) \rightarrow (p \wedge q)$ is false then the truth values of the statements $p, q, r$ respectively can be:

Options:

(A) TFT

(B) FFT

(C) TFF

(D) FTF

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the truth values of $p$, $q$, and $r$ given that the Boolean expression $((p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)) \rightarrow (p \wedge q)$ has a truth value of **False**.

Logic is not about memorizing tables; it is about systematic detective work. We must dismantle the expression step-by-step to uncover the underlying values.

### The Master Key

The entire expression is an implication of the form $A \rightarrow B$. In formal logic, an implication is only **False** when the antecedent $A$ is **True** and the consequent $B$ is **False**.

This provides us with two critical conditions:
1. The antecedent $A = (p \vee q) \wedge (q \rightarrow r) \wedge (\sim r)$ must be **True**.
2. The consequent $B = (p \wedge q)$ must be **False**.

### The Antecedent Decomposition

Since the antecedent $A$ is a conjunction of three blocks joined by the $\wedge$ (AND) operator, every single link in the chain must be **True** for the entire expression to be **True**.

This yields three micro-equations:
1. $(p \vee q) = T$
2. $(q \rightarrow r) = T$
3. $(\sim r) = T$

### The Domino Effect

We begin with the simplest piece: $(\sim r) = T$. If the negation of $r$ is **True**, then $r$ itself must be **False** ($F$).

Next, we substitute $r = F$ into the second block: $(q \rightarrow F) = T$. For an implication to be **True** when the consequent is **False**, the antecedent $q$ must also be **False** ($F$).

Finally, we address the first block: $(p \vee q) = T$. Substituting $q = F$, we get $(p \vee F) = T$. In an OR operation, if one input is **False**, the other must be **True** for the result to be **True**. Therefore, $p$ must be **True** ($T$).

### Final Verification

We have deduced the values $p=T$, $q=F$, and $r=F$. We must now verify these against our second condition: $(p \wedge q) = F$.

Substituting our values, we get:
$$ (T \wedge F) = F $$

The condition holds perfectly. The final truth values are **$p=T$**, **$q=F$**, and **$r=F$**.

Animated Solution for Mathematics - Sets and Relations: If the truth value of the Boolean expression $((p \lor q) \land (q \to r) \land (\sim r)) \to (p \land q)$ is false then the truth values of the statements $p, q, r$ respectively can be:

Select Answer:

Visualized Solution

Boolean Expression Analysis

The Implication Rule

Setting the Conditions

Analyzing the Antecedent

Solving for $r$

Solving for $q$

Solving for $p$

Verifying the Consequent

Final Truth Values

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Master Key

The Antecedent Decomposition

The Domino Effect

Final Verification

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Select Answer:

Visualized Solution

Boolean Expression Analysis

The Implication Rule

Setting the Conditions

Analyzing the Antecedent

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Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q

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