Equivalence Relation With an Example

In this class, We discuss Equivalence Relation With an Example.

The reader should have prior knowledge of transitive and reflexive relations. Click Here.

Equivalence Relation:

A relation is equivalence if it satisfies 1) Reflexive, 2) Symmetric 3) Transitive properties.

Example:

A = set of real numbers

R = {(a, b) / a- b is divisible by 5}

Relation ” R” is reflexive

Take any real number x then (x, x) present in the relation.

Take real number 1, then 1 – 1 = 0

zero divisible by five.

The relation “R” is symmetric

Take elements (x, y)

If (x, y) is present in the relation, then (y, x) will be present.

If x – y is divisible by five, then y – x is also divisible by five.

The relation R is transitive.

If xRy and yRz then xRz

Example:

(15, 10) and (10, 5)

(15, 5) also divisible by 5.

Example 2:

A = (a, b)

S = set of all subsets of the set A.

S = {Φ, {a}, {b}, {a,b}}

Relation R = [S, ⊆]

The relation R is reflexive and not symmetric.

The relation R is not an equivalence relation.