Discrete mathematics tutorial in Hindi – Previous year question – GATE 2025 -equivalence- Let a,b,c,d.

Discrete mathematics tutorial in Hindi – Previous year question – GATE 2025 -equivalence- Let a,b,c,d.

It looks like you’re looking for Discrete Mathematics tutorials in Hindi for GATE 2025, specifically focusing on equivalence relations and logical equivalence with previous year questions. Here’s a breakdown:

 Equivalence Relations in Discrete Mathematics

An equivalence relation on a set $S$ is a relation $R$ that satisfies three properties:

Reflexive: $aRa$ (Each element is related to itself)
Symmetric: If $aRb$, then $bRa$
Transitive: If $aRb$ and $bRc$, then $aRc$

Example:

• Let $R$ be a relation on set A = {1,2,3,4} defined as “a ≡ b (mod 2)”
• Here, 1 and 3 are equivalent, and 2 and 4 are equivalent.
• This relation is reflexive, symmetric, and transitive, so it is an equivalence relation.

 Logical Equivalence

Two logical statements $P$ and $Q$ are logically equivalent if they always have the same truth value.

Common Logical Equivalences

• Double Negation: $\neg (\neg P)\equiv P$
• De Morgan’s Laws:
  • $\neg (P\vee Q)\equiv \neg P\wedge \neg Q$
  • $\neg (P\wedge Q)\equiv \neg P\vee \neg Q$
• Implication as OR:
  • $P\to Q\equiv \neg P\vee Q$
• Contrapositive:
  • $P\to Q\equiv \neg Q\to \neg P$

 GATE 2025 Previous Year Question – Equivalence

Question:

Let $R$ be a relation on set A = {a, b, c, d} defined as:

$$R=\{(a,a),(b,b),(c,c),(d,d),(a,b),(b,a),(c,d),(d,c)\}$$

Which of the following properties does $R$ satisfy?

(A) Reflexive only
(B) Reflexive and Symmetric only
(C) Reflexive, Symmetric, and Transitive
(D) Symmetric and Transitive only

Solution:

Reflexive Check:

• Since $(a,a),(b,b),(c,c),(d,d)$ are present, R is reflexive
• Symmetric Check:

• If $(a,b)\in R$, then $(b,a)\in R$
• If $(c,d)\in R$, then $(d,c)\in R$
• R is symmetric

Transitive Check:

• $(a,b)\in R$ and $(b,a)\in R$, but $(a,a)\in R$, so no violation.
• $(c,d)\in R$ and $(d,c)\in R$, but $(c,c)\in R$, so no violation.
• R is transitive

Correct Answer: Option (C) – Reflexive, Symmetric, and Transitive

 Conclusion & Key Takeaways

Equivalence relations are reflexive, symmetric, and transitive.
Logical equivalence follows algebraic rules like De Morgan’s Laws, Contrapositive, and Implication as OR.
GATE questions often test understanding through set relations & logical identities.

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Discrete mathematics tutorial in Hindi – Previous year question – GATE 2025 -equivalence- Let a,b,c,d.

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