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

Let’s break down and solve a GATE-style Discrete Mathematics question related to equivalence relations and equivalence classes, as might appear in GATE 2025:

Topic: Equivalence Class and Equivalence Relation

Question (Sample Format):
Let $R$ be a relation on the set $A=\{1,2,3,4,5,6\}$ defined as:

$aRb⟺a\equiv b(\text{mod }3)$.

a) Prove that $R$ is an equivalence relation.
b) Find the equivalence classes of the set $A$ under the relation $R$.

GATE-type MCQ style follow-up:
How many equivalence classes are formed?

Step-by-step Solution

a) Check if R is an Equivalence Relation

A relation $R$ is an equivalence relation if it is:

1. Reflexive: $a\equiv a(\text{mod }3)$ – Always true

2. Symmetric: If $a\equiv b$, then $b\equiv a$ – True

3. Transitive: If $a\equiv b$ and $b\equiv c$, then $a\equiv c$ – Also true

Hence, R is an equivalence relation.

b) Find the Equivalence Classes

We calculate $a\mathrm{mod}3$ for each element in the set $A=\{1,2,3,4,5,6\}$:

So, the equivalence classes are:

• $[0]=\{3,6\}$

• $[1]=\{1,4\}$

• $[2]=\{2,5\}$

Answer: 3 equivalence classes

Final Answer for GATE-type MCQ:

How many equivalence classes are formed?
Option C: 3

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