Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic.

Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic.

 Tautology Concept – GATE 2025 (Discrete Mathematics)

 What is a Tautology?

A tautology is a propositional logic statement that is always true regardless of the truth values of its components.

Mathematical Representation:
If $P$ is a logical proposition, then $P$ is a tautology if its truth table always evaluates to “True” (T) for all possible input values.

 Example of a Tautology

1. Logical OR with a Truth Statement

   $$P\vee \neg P$$

   • This means “P OR NOT P”
   • This is always true because either $P$ is true, or $\neg P$ is true.
   • Example:
     • If $P=T$, then $\neg P=F$ → $T\vee F=T$
     • If $P=F$, then $\neg P=T$ → $F\vee T=T$
   • Always True → Tautology

2. Implication Rule as a Tautology

   $$(P\to Q)\vee (\neg P)$$

   • This means “If P implies Q, OR not P”
   • Always evaluates to True in a truth table.

 GATE 2025 Previous Year Question on Tautology

Let $P,Q,R$ be three atomic propositions. The statement is:

$$(P\vee Q)\to (P\vee Q\vee R)$$

Solution Using Truth Table

 The last column is always TRUE, so this statement is a tautology.

 Key Takeaways for GATE 2025

1. Tautology: A statement that is always true for all values of P, Q, and R.
2. Common Tautology Forms:
   • $P\vee \neg P=T$
   • $(P\to Q)\vee \neg P$
   • $P\vee Q\to P\vee Q\vee R$
3. GATE Questions often test tautology using truth tables, logical identities, and set theory.

 Would you like more solved GATE questions on Tautology for practice?

Tautology concept of one- GATE 2025 previous year gate question paper Let P,Q,R be three atomic.