Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence.

Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence.

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It looks like you are looking for Discrete Mathematics – Equivalence Relations – Previous Year GATE Questions related to GATE 2025. Here’s a conceptual breakdown along with a sample problem:

 Equivalence Relation – Definition

A relation R on a set S is called an Equivalence Relation if it satisfies the following three properties:

1. Reflexive: $aRa$ for all $a\in S$.
2. Symmetric: If $aRb$, then $bRa$ for all $a,b\in S$.
3. Transitive: If $aRb$ and $bRc$, then $aRc$ for all $a,b,c\in S$.

 Example GATE Question on Equivalence Relations

Question:
Let R and S be two equivalence relations on a set $A$. Consider the relation $R\cap S$ (intersection of R and S). Is $R\cap S$ also an equivalence relation?

Solution Approach:

• Reflexive: Since both $R$ and $S$ are reflexive, $(a,a)$ belongs to both $R$ and $S$, so it must belong to $R\cap S$.
• Symmetric: If $(a,b)\in R\cap S$, then $(a,b)\in R$ and $(a,b)\in S$. Since both $R$ and $S$ are symmetric, $(b,a)\in R$ and $(b,a)\in S$, so $(b,a)\in R\cap S$.
• Transitive: If $(a,b)\in R\cap S$ and $(b,c)\in R\cap S$, then $(a,b)$ and $(b,c)$ belong to both $R$ and $S$. Since both $R$ and $S$ are transitive, $(a,c)\in R$ and $(a,c)\in S$, so $(a,c)\in R\cap S$.
• Conclusion: Since $R\cap S$ satisfies all three properties, it is also an equivalence relation.
• Answer: Yes, $R\cap S$ is always an equivalence relation.

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Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence.

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