Discrete Mathematics Gate Question-GATE 2025 equivalence class A relation R is defined on the set

Discrete Mathematics Gate Question-GATE 2025 equivalence class A relation R is defined on the set

It looks like you’re looking for a GATE 2025 Discrete Mathematics question related to Equivalence Classes and Relations. Here’s a general breakdown to help you understand how to solve such problems.

 Equivalence Class & Relation Definition

A relation R on a set S is an equivalence relation if it satisfies three properties:
Reflexivity: $aRa$ for all $a\in S$
Symmetry: If $aRb$, then $bRa$
Transitivity: If $aRb$ and $bRc$, then $aRc$

If a relation R is an equivalence relation, then the equivalence class of an element a is the set:

$$[a]=\{x\in S\mid xRa\}$$

This means all elements in an equivalence class are related to each other under R.

 Example GATE Question:

Let $S=\{1,2,3,4,5,6\}$ and define a relation R as:

$$aRb⟺a\equiv b(\mathrm{mod}3)$$

(i.e., a is related to b if they have the same remainder when divided by 3).

 Find the Equivalence Classes:

We check numbers modulo 3:

• [1] = {1, 4} (since $1\mathrm{mod}3=1$ and $4\mathrm{mod}3=1$)
• [2] = {2, 5} (since $2\mathrm{mod}3=2$ and $5\mathrm{mod}3=2$)
• [3] = {3, 6} (since $3\mathrm{mod}3=0$ and $6\mathrm{mod}3=0$)

Final Answer: The equivalence classes are {1, 4}, {2, 5}, {3, 6}.

 How to Solve Similar GATE Questions?

Check Reflexivity, Symmetry, and Transitivity to confirm R is an equivalence relation.
Find elements that are related to each other based on the given rule.
Group elements into disjoint equivalence classes.

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Discrete Mathematics Gate Question-GATE 2025 equivalence class A relation R is defined on the set

Discrete Mathematics for Computer Science