What Is Equivalence Class in Relations and Functions Class 12: Definition & Examples

Definition of Equivalence Class in Relations and Functions

An equivalence class is a subset formed by all elements related to a particular element under an equivalence relation.

In Class 12 NCERT Mathematics, an equivalence relation $R$ on a set $A$ satisfies three properties:

• Reflexive: For every $a \in A$, $(a, a) \in R$
• Symmetric: If $(a, b) \in R$, then $(b, a) \in R$
• Transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$

For an element $a \in A$, the equivalence class of $a$ is denoted by $[a]$ and defined as:

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

This means $[a]$ contains all elements related to $a$ under $R$.

Properties of Equivalence Classes

Equivalence classes have important properties that help in understanding relations:

• Partitioning: The set $A$ is partitioned into disjoint equivalence classes. Every element of $A$ belongs to exactly one equivalence class.
• Non-overlapping: If $[a]$ and $[b]$ are two equivalence classes, then either $[a] = [b]$ or $[a] \cap [b] = \varnothing$.
• Representative Element: Each equivalence class can be represented by any of its members.

Example: Consider the set $A = \{1, 2, 3, 4\}$ with relation $R$ defined by $aRb$ if $a \equiv b \pmod{2}$. Here:

• $[1] = \{1, 3\}$ (odd numbers)
• $[2] = \{2, 4\}$ (even numbers)

These two classes partition $A$ with no overlap.

Equivalence Relation vs Other Relations: A Comparison

Understanding how equivalence relations differ from other types of relations is key for Class 12 students.

Only when all three properties hold simultaneously, a relation is an equivalence relation, which then forms equivalence classes.

How to Find Equivalence Classes: Step-by-Step Example

To find equivalence classes, follow these steps:

1. Identify the relation $R$ and set $A$. 2. Check if $R$ is an equivalence relation by verifying reflexivity, symmetry, and transitivity. 3. Pick an element $a$ in $A$. 4. Find all elements related to $a$ under $R$. 5. Form the equivalence class $[a]$ with these elements. 6. Repeat for other elements not yet assigned to any class.

Worked Example:

Set $A = \{1, 2, 3, 4, 5\}$, relation $R$ defined by $aRb$ if $a - b$ is divisible by 3.

• Check properties: $R$ is reflexive, symmetric, transitive.
• Find $[1]$: Elements where $1 - x$ divisible by 3 are $1$ and $4$.
• Find $[2]$: Elements where $2 - x$ divisible by 3 are $2$ and $5$.
• $[3] = \{3\}$ alone.

Thus, the equivalence classes are:

$$ [1] = \{1, 4\}, \quad [2] = \{2, 5\}, \quad [3] = \{3\} $$

Importance of Equivalence Classes in Class 12 Mathematics

Equivalence classes simplify the study of relations and functions by grouping elements with similar properties.

• They help in partitioning sets into meaningful subsets.
• Used in defining quotient sets and modular arithmetic.
• Aid in understanding functions, especially when dealing with injective, surjective, and bijective functions.
• Essential for solving problems in linear algebra, number theory, and other advanced topics.

Mastering equivalence classes is crucial for CBSE Class 12 exams and deeper mathematical understanding.

Summary and Tips to Remember Equivalence Classes

To excel in the topic "what is equivalence class in relations and functions class 12," remember:

• Equivalence classes arise only from equivalence relations.
• Each class contains elements related to one another.
• Classes partition the original set without overlap.
• Use examples like modular arithmetic to visualize classes.
• Practice NCERT exercises and solve related problems.

Consistent practice will build confidence for your Class 12 exams.