Equivalence Class are the group of elements of a set based on a specific notion of equivalence defined by an equivalence relation. An equivalence relation is a relation that satisfies three properties: reflexivity, symmetry, and transitivity. Equivalence classes partition the set S into disjoint subsets. Each subset consists of elements that are related to each other under the given equivalence relation.
In this article, we will discuss the concept of Equivalence Class in sufficient detail including its definition, example, properties, as well as solved examples.
What are Equivalence Classes?
An equivalence class is the name that we give to the subset of S which includes all elements that are equivalent to each other. “Equivalent” is dependent on a specified relationship, called an equivalence relation. If there’s an equivalence relation between any two elements, they’re called equivalent.
Equivalence Class Definition
Given an equivalence relation on a set S, an equivalence class with respect to an element a in S is the set of all elements in S that are related to a i.e.,
[a] OR {x ϵ S| x is related to a}
For example, consider the set of integers ℤ and the equivalence relation defined by congruence modulo n. Two integers a and b are considered equivalent (denoted as (a ≡ b mod(n) if they have the same remainder when divided by n. In this case, the equivalence class of an integer a is the set of all integers that have the same remainder as a when divided by n.
What is Equivalence Relation?
Any relation R, is said to be Equivalence Realtion if and only if, it satisfy the following three condition:
- Reflexivity: For any element a, a is related to itself.
- Symmetry: If a is related to b, then b is related to a.
- Transitivity: If a is related to b, and b is related to c, then a is related to c.
Read more about Equivalence Relation.
Some examples of equivalence relation are:
Equality on a Set: Let X be any set, and define a relation R on X such that a R b if and only if a = b for a, b ϵ X.
- Reflexivity: For every a ϵ X, a = a (trivially true).
- Symmetry: If a = b, then b = a (trivially true).
- Transitivity: If a = b and b = c, then a = c (trivially true).
Congruence modulo n: Let n be a positive integer, and define a relation R on the integers ℤ such that a R b if and only if a – b is divisible by n.
- Reflexivity: For every a ϵ ℤ, a – a = 0 is divisible by n.
- Symmetry: If a – b is divisible by n, then -(a – b) = b – a is also divisible by n.
- Transitivity: If a – b is divisible by n and b – c is divisible by n, then a – c is also divisible by n.
Examples of Equivalence Class
The well-known example of an equivalence relation is the “equal to (=)” relation. In other words, two elements of the given set are equivalent to each other if they belong to the same equivalence class. The equivalence relationships can be explained in terms of the following examples:
Equivalence Relation on Integers
Equivalence Relation: Congruence modulo 5 (a ≡ b [mod(5)] )
- Equivalence class of 0: [0] = {. . ., -10, -5, 0, 5, 10, . . .}
- Equivalence class of 1: [1] = {. . ., -9, -4, 1, 6, 11, . . .}
- Equivalence class of 2: [2] = {. . ., -8, -3, 2, 7, 12, . . .}
- Equivalence class of 3: [3] = {. . ., -7, -2, 3, 8, 13, . . .}
- Equivalence class of 4: [4] = {. . ., -6, -1, 4, 9, 14, . . .}
Equivalence Relation on Real Numbers
Equivalence Relation: Absolute difference (a ~ b if |a – b| < 1)
- Equivalence class of 0: [0] = (-0.5, 0.5)
- Equivalence class of 1: [1] = (0.5, 1.5)
- Equivalence class of 2: [2] = (1.5, 2.5)
- Equivalence class of 3: [3] = (2.5, 3.5)
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Properties of Equivalence Classes
The properties of equivalence classes are:
- Every element belongs to exactly one equivalence class.
- Equivalence classes are disjoint i.e., intersection of any two equivalence class is null set.
- The union of all equivalence classes is the original set.
- Two elements are equivalent if and only if their equivalence classes are equal.
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Equivalence Classes and Partition
Groups of elements in a set related by an equivalence relation, whereas a collection of these equivalence classes, covering the entire set with no overlaps are called partition.
Difference between Equilavalence Classes and Partition
The key difference between Equilavalence Classes and Partition are given in the following table:
| Sets of elements that are considered equivalent under a relation. |
A collection of non-empty, pairwise disjoint subsets such that their union is the entire set. |
| If A is an equivalence class, it is often denoted as [a] or [a]R, where a is a representative element and R is the equivalence relation. |
A partition of a set X is denoted as {B1, B2, … ,Bn}, where Bi are the disjoint subsets in the partition. |
| Equivalence classes form a partition of the underlying set. |
A partition may or may not arise from an equivalence relation. |
| Equivalence classes may have different cardinalities. |
All subsets in the partition have the same cardinality. |
|
Consider the set of integers and the equivalence relation “having the same remainder when divided by 5.”
Equivalence classes are {…,−5,0,5,…}, {…,−5,0,5,…}, {…,−4,1,6,…}, and {…,−4,1,6,…}, etc.
|
Consider the set of integers partitioned into even and odd numbers:
{…,−4,−2,0,2,4,…}, and {…,−3,−1,1,3,5,…}.
|
| Equivalence classes are either disjoint or identical. |
Partitions consist of disjoint subsets. |
Solved Examples on Equivalence Class
Example 1: Prove that the relation R is an equivalence type in the set P= { 3, 4, 5,6 } given by the relation R = { (p, q):|p-q| is even}.
Solution:
Given: R = { (p, q):|p-q| is even }. Where p, q belongs to P.
Reflexive Property
From the provided relation |p – p| = | 0 |=0.
- And 0 is always even.
- Therefore, |p – p| is even.
- Hence, (p, p) relates to R
So R is Reflexive.
Symmetric Property
From the given relation |p – q| = |q – p|.
- We know that |p – q| = |-(q – p)|= |q – p|
- Hence |p – q| is even.
- Next |q – p| is also even.
- Accordingly, if (p, q) ∈ R, then (q, p) also belongs to R.
Therefore R is symmetric.
Transitive Property
- If |p – q| is even, then (p-q) is even.
- Similarly, if |q-r| is even, then (q-r) is also even.
- The summation of even numbers is too even.
- So, we can address it as p – q+ q-r is even.
- Next, p – r is further even.
Accordingly,
- |p – q| and |q-r| is even, then |p – r| is even.
- Consequently, if (p, q) ∈ R and (q, r) ∈ R, then (p, r) also refers to R.
Therefore R is transitive.
Example 2: Consider A = {2, 3, 4, 5} and R = {(5, 5), (5, 3), (2, 2), (2, 4), (3, 5), (3, 3), (4, 2), (4, 4)}.
Solution:
Given: A = {2, 3, 4, 5} and
Relation R = {(5, 5), (5, 3), (2, 2), (2, 4), (3, 5), (3, 3), (4, 2), (4, 4)}.
For R to be Equivalence Relation, R needs to satisfy three properteis i.e., Reflexive, Symmetric, and Transitive.
Reflexive: Relation R is reflexive because (5, 5), (2, 2), (3, 3) and (4, 4) ∈ R.
Symmetric: Relation R is symmetric as whenever (a, b) ∈ R, (b, a) also relates to R i.e., (3, 5) ∈ R ⟹ (5, 3) ∈ R.
Transitive: Relation R is transitive as whenever (a, b) and (b, c) relate to R, (a, c) also relates to R i.e., (3, 5) ∈ R and (5, 3) ∈ R ⟹ (3, 3) ∈ R.
Accordingly, R is reflexive, symmetric and transitive.
So, R is an Equivalence Relation.
Practice Problems on Equivalence Class
Problem 1: aRb if a+b is even. Determine if it’s an equivalence relation and its properties.
Problem 2: xSy if x and y have the same birth month. Analyze if it’s an equivalence relation.
Problem 3: Consider A = {2, 3, 4, 5} and R = {(5, 5), (5, 3), (2, 2), (2, 4), (3, 5), (3, 3), (4, 2), (4, 4)}. Confirm that R is an equivalence type of relation.
Problem 4: Prove that the relation R is an equivalence type in the set P= { 3, 4, 5,6 } given by the relation R = { (p, q):|p-q| is even }.
Equivalence Class: FAQs
1. What is the Equivalence Class?
An equivalence class is a subset within a set, formed by grouping all elements that are equivalent to each other under a given equivalence relation. It represents all members that are considered equal by that relation.
2. What is the Symbol for Equivalence Class?
The symbol for an equivalence class is typically written as [a], where “a” is a representative element of the class. This notation denotes the set of all elements equivalent to “a” under a specific equivalence relation.
3. How do you find the Equivalence Class of a Set?
To find the equivalence class of a set, follow these steps:
Step 1: Define an Equivalence Relation.
Step 2: Select an Element from Set.
Step 3: Identify Equivalent Elements to the Selected Elements.
Step 4: Form the Equivalence Class containing all the elements equivalent to the selected element.
4. What is the difference between Equivalence Class and Partition?
Equivalence classes are subsets formed by an equivalence relation, while partitions are non-overlapping subsets covering the entire set. Every equivalence class is a subset in a partition, but not every partition arises from an equivalence relation.
5. What is an Equivalence Relation?
A relation that is reflexive, symmetric, and transitive, dividing a set into disjoint subsets.
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