Let the operations *, ⊙ ∈ {∧, ∨}. If (p * q) ⊙ (p ⊙ ~ q) is a tautology, then the ordered pair (*, ⊙

Q: Wait, why do we reject the truth table method? Isn't it safer?

Truth tables are the 'brute force' of logic. While they are foolproof, they are time-consuming. In a competitive exam like JEE Advanced, time is your most precious currency. Learning to manipulate logical expressions algebraically is a superpower that saves time and reduces calculation errors. Think of it as choosing a scalpel over a sledgehammer.

Q: What if I picked the wrong option first? Does that mean I failed?

Absolutely not! In mathematics, especially in multiple-choice logic problems, testing an incorrect option is not a failure; it is a process of elimination. Every 'wrong' path you rule out brings you closer to the truth. Don't fear the wrong path; just be systematic and keep your cool.

Q: Why does $p \vee T$ always equal $T$? I keep forgetting this rule.

Think of the 'or' operator ($\vee$) as a 'filter' that lets truth pass through. If you have a statement $p$ and you 'or' it with a guaranteed truth ($T$), the result is always true because the 'or' condition is satisfied by the $T$ component. It doesn't matter if $p$ is true or false; the $T$ carries the weight!

Question

Let the operations $*, \odot \in \{\wedge, \vee\}$. If $(p * q) \odot (p \odot \sim q)$ is a tautology, then the ordered pair $(*, \odot)$ is:

Options:

(A) $(\vee, \wedge)$

(B) $(\vee, \vee)$

(C) $(\wedge, \wedge)$

(D) $(\wedge, \vee)$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Logic is the language of the universe, and tautologies are its most elegant sentences—statements that are true by their very structure, independent of the facts they contain. When we look at the expression $(p * q) \odot (p \odot \sim q)$, we are looking at a puzzle of connectivity.

Our mission is to find the operators that transform this expression into a universal truth.

### The Strategy of the Master

Many students, upon seeing this, immediately reach for the truth table. They start drawing rows and columns, sweating over every entry. But you are not 'many students.' You are a JEE aspirant.

You know that time is your most valuable asset. Instead of brute-forcing, we use the 'Strategy of Systematic Testing.' We look at our options, and we test them with the elegance of algebra. Let us test Option (B), where both operators are the disjunction, or 'or' ($\vee$).

### The Algebraic Dance

When we substitute the operators, our expression becomes:

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

Now, look closely. Because the operator is the same throughout, we are granted the freedom of the Associative Law. We can drop the parentheses entirely. It becomes $p \vee q \vee p \vee \sim q$.

This is where the magic happens. We use the Commutative Law to rearrange our terms, bringing the $p$'s together and the $q$'s together:

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

### The Collapse of Complexity

Now, we apply the laws of logic like a master craftsman. First, the Idempotent Law: $p \vee p \equiv p$. The first part of our expression simplifies instantly.

Next, we face the beautiful Law of Excluded Middle: $q \vee \sim q \equiv T$. This is the moment of truth. A statement $q$ must either be true or false; therefore, the disjunction of $q$ and its negation is always true.

Our expression has now collapsed into $p \vee T$. Finally, we invoke the Identity Law. In the realm of 'or', if one side is already true, the entire statement is true. Thus:

$$ p \vee T \equiv T $$

We have arrived at the destination. The expression is a **tautology**. You see, logic is not about memorizing tables; it is about seeing the structure, simplifying the chaos, and finding the truth hidden in plain sight.

Animated Solution for Mathematics - Sets and Relations: Let the operations $, ⊙ \in {\land, \lor}$ . If $(p q) ⊙ (p ⊙ \sim q)$ is a tautology, then the ordered pair $(*, ⊙)$ is:

Select Answer:

Visualized Solution

Identifying the Goal

Understanding Tautology

Strategy: Testing Options

Substituting Option B

Removing Parentheses

Rearranging Terms

Simplifying p terms

Simplifying q terms

Final Tautology Verification

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Strategy of the Master

The Algebraic Dance

The Collapse of Complexity

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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; } Let the operations ∗,⊙∈{∧,∨}. If (p∗q)⊙(p⊙∼q) is a tautology, then the ordered pair (∗,⊙) is: