If P and Q are two statements, then which of the following compound statement is a tautology? (A) ((

Q: Wait, why do we bother simplifying the expression instead of just drawing a truth table?

Truth tables are a safe harbor, but in the heat of the JEE Advanced exam, they are time-consuming and prone to silly errors. Algebraic simplification using laws like De Morgan's or Distributive laws is the 'sniper' approach—it is faster, more elegant, and builds a deeper intuition for how logical structures interact.

Q: I always get confused with De Morgan's Law when there is a negation outside a bracket. How do I handle it?

Think of the negation symbol $\sim$ as a 'distributor' that flips everything it touches. When it enters a parenthesis, it negates the statements and flips the operator: $\wedge$ becomes $\vee$, and $\vee$ becomes $\wedge$. So, $\sim(A \wedge B)$ becomes $\sim A \vee \sim B$. Practice this until it becomes muscle memory!

Q: Why is $P \vee \sim P$ always a tautology?

This is the 'Law of Excluded Middle.' In classical logic, a statement must be either true or false. If $P$ is true, then $\sim P$ is false, and $T \vee F$ is true. If $P$ is false, then $\sim P$ is true, and $F \vee T$ is true. Since it covers all possibilities, it is always true!

Q: Is there a physical way to visualize these logical gates?

Absolutely! Think of $\wedge$ as a series circuit (both switches must be closed for current to flow) and $\vee$ as a parallel circuit (current flows if either switch is closed). A tautology is like a circuit that is permanently wired to be 'on' regardless of how you toggle the switches.

Q: How do I know which option to test first?

In multiple-choice questions, look for the most 'logical' conclusion. The antecedent $((P \Rightarrow Q) \wedge \sim Q)$ is essentially the structure of 'Modus Tollens'—a classic rule of inference. If $P$ implies $Q$, and $Q$ is false, then $P$ must be false. This points directly to $\sim P$ being the likely conclusion.

Question

If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?

Options:

(A) $((P \Rightarrow Q) \wedge \sim Q) \Rightarrow Q$

(B) $((P \Rightarrow Q) \wedge \sim Q) \Rightarrow \sim P$

(C) $((P \Rightarrow Q) \wedge \sim Q) \Rightarrow P$

(D) $((P \Rightarrow Q) \wedge \sim Q) \Rightarrow (P \wedge Q)$

Sigma Percentile · Accepted Answer

### The Architecture of Truth: Decoding Logical Tautologies

Welcome, future engineer. Today, we are not just solving a problem; we are peeling back the layers of human reasoning itself.

In the world of JEE Advanced, logic is not just a chapter—it is the foundation upon which all your physics and mathematics are built. When we look at a compound statement like $((P \Rightarrow Q) \wedge \sim Q) \Rightarrow \sim P$, we are looking at the very structure of a valid argument.

### Phase 1: The Anatomy of the Antecedent

Let us define our core expression as $L = ((P \Rightarrow Q) \wedge \sim Q)$. Before we can determine if the entire implication is a tautology, we must understand what $L$ represents.

To prove the structure, we use the identity $P \Rightarrow Q \equiv \sim P \vee Q$. Substituting this into our expression, we get:

$$L \equiv (\sim P \vee Q) \wedge \sim Q$$

Now, we apply the Distributive Law: $(A \vee B) \wedge C \equiv (A \wedge C) \vee (B \wedge C)$. This transforms our expression into:

$$L \equiv (\sim P \wedge \sim Q) \vee (Q \wedge \sim Q)$$

### Phase 2: The Beauty of Contradiction

Look closely at the term $(Q \wedge \sim Q)$. This is a logical contradiction—a statement cannot be both true and false simultaneously.

Therefore, $Q \wedge \sim Q \equiv F$. Our expression now simplifies to:

$$L \equiv (\sim P \wedge \sim Q) \vee F$$

Since any statement $X$ OR-ed with False remains $X$ (the Identity Law), we arrive at the conclusion:

$$L \equiv \sim P \wedge \sim Q$$

### Phase 3: The Final Verification

Now, we test the full implication: $L \Rightarrow \sim P$. Substituting our simplified $L$, we get:

$$(\sim P \wedge \sim Q) \Rightarrow \sim P$$

To evaluate this, we use the implication rule $A \Rightarrow B \equiv \sim A \vee B$ once more. This yields:

$$\sim (\sim P \wedge \sim Q) \vee \sim P$$

Apply De Morgan's Law to the first part, where $\sim (\sim P \wedge \sim Q) \equiv P \vee Q$. Our expression is now:

$$(P \vee Q) \vee \sim P$$

### The Grand Finale

By the Associative Law, we can rearrange the terms to group the complementary variables:

$$(P \vee \sim P) \vee Q$$

We have reached the **Law of Excluded Middle**, where $P \vee \sim P \equiv T$. Our expression becomes:

$$T \vee Q$$

In the realm of logic, any statement OR-ed with a Tautology is always a Tautology. Thus, $T \vee Q \equiv T$.

We have successfully proven that the statement is **always true**, regardless of the truth values of $P$ and $Q$.

Animated Solution for Mathematics - Sets and Relations: If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?

Select Answer:

Visualized Solution

Defining the Core Expression $L$

Converting Implication to Disjunction

Applying the Distributive Law

Identifying the Contradiction

Final Simplified Form of $L$

Testing Option Two

Converting the Implication Again

Applying De Morgan's Law

Using the Associative Law

The Law of Excluded Middle

Final Verification of Tautology

Conclusion

The Sigma Insight: Types of Sets and Set Operations

The Architecture of Truth

Phase 1

Phase 2

Phase 3

The Grand Finale

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Animated Solution for Mathematics - Sets and Relations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P and Q are two statements, then which of the following compound statement is a tautology?

Select Answer:

Visualized Solution

Defining the Core Expression .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L

Converting Implication to Disjunction

Applying the Distributive Law

Identifying the Contradiction

Final Simplified Form of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L

Testing Option Two

Converting the Implication Again

Applying De Morgan's Law

Using the Associative Law

The Law of Excluded Middle

Final Verification of Tautology

Conclusion

The Sigma Insight: Types of Sets and Set Operations

The Architecture of Truth

Phase 1

Phase 2

Phase 3

The Grand Finale

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Animated Solution for Mathematics - Sets and Relations: If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?

Defining the Core Expression $L$

Final Simplified Form of $L$

If $q$ is false and $p \land q \leftrightarrow r$ is true, then which one of the following statements is a tautology?

Among the statements $(S 1) : (p \Rightarrow q) \lor ((\sim p) \land q)$ is a tautology, $(S 2) : (q \Rightarrow p) \Rightarrow ((\sim p) \land q)$ is a contradiction