If the Boolean expression (p ∧ q) ⊛ (p ⊗ q) is a tautology, then ⊛ and ⊗ are respectively given by (

Q: Why do we test options instead of solving it algebraically?

In competitive exams like JEE, time is your most precious resource. While you could derive the operators from scratch, testing the options is often faster. It turns an abstract search into a concrete verification process, which is less prone to error.

Q: What if I don't remember the Implication Law?

If you forget the Implication Law, you can always fall back on a truth table. However, truth tables can be tedious and time-consuming. The Implication Law is a powerful shortcut that simplifies complex logical expressions into basic OR and AND operations, making the path to the solution much clearer.

Q: Why does the expression simplify to a tautology?

A tautology is a statement that is always true. In our simplification, we reached the form $\neg p \vee (\neg q \vee q)$. Since $\neg q \vee q$ is always true (Complement Law), the entire expression becomes $\neg p \vee T$, which is always true regardless of the value of $p$ (Identity Law).

Question

If the Boolean expression $(p \wedge q) \circledast (p \otimes q)$ is a tautology, then $\circledast$ and $\otimes$ are respectively given by

Options:

(A) $\rightarrow, \rightarrow$

(B) $\wedge, \vee$

(C) $\vee, \rightarrow$

(D) $\wedge, \rightarrow$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The objective is to determine the operators $\circledast$ and $\otimes$ such that the expression $(p \wedge q) \circledast (p \otimes q)$ results in a **tautology**. A tautology is a statement that remains true for all possible truth values of the variables $p$ and $q$.

To solve this efficiently, we test the hypothesis where $\circledast = ightarrow$ and $\otimes = ightarrow$. This substitution yields the logical expression:

$$(p \wedge q) ightarrow (p ightarrow q)$$

### The Toolkit

To dismantle this expression, we utilize the **Implication Law**, which states that $A ightarrow B \equiv 
eg A \vee B$. This law is essential as it converts rigid implications into the flexible domain of disjunctions and negations.

Applying this to the outer implication, we obtain:

$$
eg(p \wedge q) \vee (p ightarrow q)$$

Next, we address the inner bracket. Applying the same law to $p ightarrow q$, we get $
eg p \vee q$. The expression now reads:

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

### The Simplification

We now apply **De Morgan's Law** to the first term. Distributing the negation, we find that $
eg(p \wedge q) \equiv 
eg p \vee 
eg q$.

Substituting this back into our expression, we get:

$$(
eg p \vee 
eg q) \vee (
eg p \vee q)$$

Since every operator is now a disjunction ($\vee$), the **Associative Law** allows us to remove the brackets and regroup the terms:

$$(
eg p \vee 
eg p) \vee (
eg q \vee q)$$

### The Elegance of Cancellation

We now apply the fundamental laws of Boolean algebra to simplify the expression. According to the **Idempotent Law**, $
eg p \vee 
eg p \equiv 
eg p$.

Simultaneously, the **Complement Law** dictates that $
eg q \vee q \equiv T$ (where $T$ represents True). The expression simplifies to:

$$
eg p \vee T$$

Finally, the **Identity Law** states that any statement ORed with True is always True. Therefore, the entire expression simplifies to $T$.

**Conclusion:** The operators are both implication arrows ($ightarrow$), and the statement is confirmed to be a **tautology**.

Animated Solution for Mathematics - Sets and Relations: If the Boolean expression $(p \land q) ⊛ (p \otimes q)$ is a tautology, then $⊛$ and $\otimes$ are respectively given by

Select Answer:

Visualized Solution

Understanding the Problem

Testing Option 1

Applying Implication Law

Simplifying the Inner Bracket

Applying De Morgan's Law

Using Associative Law

Applying Idempotent Law

Applying Complement Law

Final Simplification to Tautology

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Toolkit

The Simplification

The Elegance of Cancellation

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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 the Boolean expression (p∧q)⊛(p⊗q) is a tautology, then ⊛ and ⊗ are respectively given by