Implication Concept Discrete mathematics previous year paper – GATE 2025- The following resolution.

Implication Concept Discrete mathematics previous year paper – GATE 2025- The following resolution.

 Implication Concept in Discrete Mathematics – GATE 2025

 What is an Implication?

In propositional logic, an implication is a logical statement of the form:

$$P\to Q$$

This means “If $P$ is true, then $Q$ must also be true.”

Truth Table for Implication ($P\to Q$)

Key Observations:

1. If $P$ is true and $Q$ is false, then $P\to Q$ is false.
2. In all other cases, $P\to Q$ is true.
3. When $P$ is false, the implication $P\to Q$ is always true, regardless of $Q$.

 Important Properties of Implication

Contrapositive:

$$P\to Q\equiv \neg Q\to \neg P$$

(Logically equivalent, used in proofs)

Inverse:

$$\neg P\to \neg Q$$

(Not logically equivalent to $P\to Q$)

Converse:

$$Q\to P$$

(Not logically equivalent to $P\to Q$)

Material Implication:

$$P\to Q\equiv \neg P\vee Q$$

(Can be rewritten in terms of OR operation)

 GATE 2025 Previous Year Question on Implication

Question:

Given three propositions $P,Q,R$, which of the following is a valid logical equivalence?

(A) $(P\to Q)\vee R\equiv (P\vee R)\to (Q\vee R)$
(B) $(P\to Q)\vee R\equiv (P\to (Q\vee R))$
(C) $(P\to Q)\vee R\equiv ((P\vee R)\to Q)$
(D) $(P\to Q)\vee R\equiv (\neg P\vee (Q\vee R))$

Solution Approach:

We use Material Implication:

$$P\to Q\equiv \neg P\vee Q$$

Now, rewriting option (D):

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

Correct Answer: Option (D)

 Conclusion

Implication ($P\to Q$) is False only when $P$ is True and $Q$ is False.
Logical Equivalences:

• $P\to Q\equiv \neg P\vee Q$
• $P\to Q\equiv \neg Q\to \neg P$ (Contrapositive)
  GATE questions often use truth tables and logical identities.

 Want more solved GATE questions on Implication?

Implication Concept Discrete mathematics previous year paper – GATE 2025- The following resolution.

Discrete Mathematics for Computer Science