Here is a clear, simple, and exam-oriented explanation of Equivalence Relations with examples. Perfect for BCA/MCA/University exams.

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⭐ Equivalence Relations

An equivalence relation on a set A is a relation that satisfies all three main properties:

1. Reflexive
2. Symmetric
3. Transitive

If a relation has all three properties, then it groups elements into equivalence classes that behave like “similarity groups.”

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⭐ Formal Definition

Let ( R \subseteq A \times A ) be a relation on set A.

R is an equivalence relation if for all ( a, b, c \in A ):

1. Reflexive:
   [
   (a,a) \in R
   ]
2. Symmetric:
   [
   (a,b) \in R \Rightarrow (b,a) \in R
   ]
3. Transitive:
   [
   (a,b) \in R \land (b,c) \in R \Rightarrow (a,c) \in R
   ]

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⭐ Examples of Equivalence Relations

✔ Example 1: Equality Relation (=)

On real numbers:

• Reflexive: a = a
• Symmetric: if a = b → b = a
• Transitive: if a = b and b = c → a = c

So “=” is an equivalence relation.

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✔ Example 2: Congruence Modulo n

Define relation on integers:

[
a R b \iff (a – b) \text{ is divisible by } n
]

For example, mod 3:

• 4 ≡ 1 (mod 3)
• 7 ≡ 1 (mod 3)

This relation is:

• Reflexive: a – a = 0 is divisible by 3
• Symmetric: if (a – b) divisible → (b – a) divisible
• Transitive: if (a – b) & (b – c) divisible → (a – c) divisible

So it is an equivalence relation.

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✔ Example 3: “Same Date of Birth” Relation

On set of students:

• Reflexive: every student has the same DOB as himself
• Symmetric: if A has same DOB as B → B has same DOB as A
• Transitive: if A and B same DOB, and B and C same DOB → A and C same DOB

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✔ Example 4: “Congruent Triangles”

Two triangles are related if they are congruent.

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⭐ Non-examples (Not Equivalence)

1. Less than (<)
   • Not reflexive
   • Not symmetric
   • Transitive ✔
     → NOT equivalence
2. Divides (|)
   • Reflexive ✔
   • Symmetric ✘
   • Transitive ✔
     → NOT equivalence

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⭐ Equivalence Class

For an equivalence relation R on set A:

The equivalence class of an element a is:

[
[a] = {x \in A \mid (a,x) \in R}
]

Meaning: the set of all elements related to a.

Example (mod 3):

A = integers
Relation: a ≡ b (mod 3)

Equivalence class of 1:

[
[1] = {1, 4, 7, 10, 13, \ldots}
]

All numbers giving remainder 1 when divided by 3.

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⭐ Partition of a Set

An equivalence relation divides a set into disjoint groups (blocks), called a partition.

Every element of the set belongs to exactly one equivalence class.

Example (mod 3):

Integers divided into three classes:

• [0] = {0,3,6,9,…}
• [1] = {1,4,7,…}
• [2] = {2,5,8,…}

These sets are disjoint & cover the whole set.

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⭐ Key Theorem

Every equivalence relation corresponds to a partition, and every partition defines an equivalence relation.

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⭐ Quick Summary for Exam

A relation R is an equivalence relation if it is:

Property	Meaning
Reflexive	(a, a) ∈ R
Symmetric	(a,b) ∈ R ⇒ (b,a) ∈ R
Transitive	(a,b),(b,c) ∈ R ⇒ (a,c) ∈ R

✔ Produces equivalence classes
✔ Creates partitions of the set
✔ Model of “similarity relations”

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