The negation of the Boolean expression p ∨ (~ p ∧ q) is equivalent to : (A) ~ p ∨ q (B) p ∧ ~ q (C)

Q: Wait, why does the OR symbol $\vee$ change to an AND symbol $\wedge$ when we negate the expression?

That is the essence of De Morgan's Law! Think of it this way: if it is NOT the case that (A OR B) is true, then it must be the case that both A is false AND B is false. Therefore, the negation of an OR operation is the AND of the individual negations. It is a fundamental rule of logical symmetry.

Q: Can I just use a truth table to solve this instead of all this algebra?

Absolutely! A truth table is a fantastic, foolproof way to verify your answer. However, mastering the algebraic manipulation is a superpower for JEE Advanced. It is faster, less prone to transcription errors, and builds a deeper intuition for how logical structures transform, which is essential for more complex problems.

Q: What happens to the 'False' (F) term in the expression $F \vee (\sim p \wedge \sim q)$?

That is the Identity Law in action. In Boolean algebra, False acts like the number zero in addition. Just as $0 + x = x$, combining any statement with False using an OR operation leaves the statement unchanged. So, $F \vee X$ simply becomes $X$. The False term effectively vanishes!

Question

The negation of the Boolean expression $p \vee (\sim p \wedge q)$ is equivalent to :

Options:

(A) $\sim p \vee q$

(B) $p \wedge \sim q$

(C) $\sim p \vee \sim q$

(D) $\sim p \wedge \sim q$

Sigma Percentile · Accepted Answer

### The Architecture of Thought: Unlocking Boolean Logic

Welcome, future engineer! Today, we are not just solving a problem; we are peeling back the layers of a logical statement to see the elegant, crystalline structure underneath.

Boolean algebra is the language of the digital world—it is the silent code running through the processor of the device you are using right now. Let us master it.

### Phase 1: The Outer Shell

We start with the expression: $p \vee (\sim p \wedge q)$. Our mission is to find its negation.

Imagine this expression as a fortress. To negate it, we must place a negation operator outside the entire structure:

$$\sim [p \vee (\sim p \wedge q)]$$

This is our starting point. We are looking for the logical opposite of this entire configuration.

### Phase 2: The De Morgan Breakthrough

How do we break into this fortress? We use De Morgan's Law. Think of it as the distributive property of logic.

When we apply the negation to the outer bracket, the OR operator ($\vee$) flips into an AND operator ($\wedge$). This gives us:

$$\sim p \wedge \sim (\sim p \wedge q)$$

We have successfully breached the outer wall!

### Phase 3: The Inner Sanctum

Now, look at the second part: $\sim (\sim p \wedge q)$. We need to apply De Morgan's Law again, but this time to the inner AND operation.

The AND ($\wedge$) flips to an OR ($\vee$). Simultaneously, we encounter a double negation on $p$. Just as two negatives make a positive in arithmetic, $\sim (\sim p)$ simplifies beautifully to $p$.

Our expression now stands as:

$$\sim p \wedge (p \vee \sim q)$$

### Phase 4: The Distributive Dance

We are almost there. Now, we use the Distributive Law, which is identical to expanding brackets in standard algebra.

We distribute the $\sim p \wedge$ across the terms inside the parenthesis $(p \vee \sim q)$. This yields:

$$(\sim p \wedge p) \vee (\sim p \wedge \sim q)$$

### Phase 5: The Vanishing Act

Look closely at the first term: $(\sim p \wedge p)$. This is a classic contradiction.

A statement and its own negation cannot both be true simultaneously. This is the Complement Law, and it tells us that $(\sim p \wedge p)$ is always False ($F$). Our expression simplifies to:

$$F \vee (\sim p \wedge \sim q)$$

Finally, we apply the Identity Law. Just as adding zero to a number changes nothing, OR-ing a statement with False changes nothing.

The False term vanishes, leaving us with our final, elegant result:

$$\sim p \wedge \sim q$$

This matches option four perfectly. You have navigated the logic, simplified the complexity, and arrived at the truth. Keep this clarity with you as you tackle the rest of your journey!

Animated Solution for Mathematics - Sets and Relations: The negation of the Boolean expression $p \lor (\sim p \land q)$ is equivalent to :

Select Answer:

Visualized Solution

Identify the Goal

De Morgan's Law (Outer)

De Morgan's \& Double Negation

Distributive Law

Complement Law

Identity Law \& Conclusion

The Sigma Insight: Types of Sets and Set Operations

The Architecture of Thought

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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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; } The negation of the Boolean expression p∨(∼p∧q) is equivalent to :