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.

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.