# Equivalence Relation

**DEFINITION1:** Let be a set and . If the relation is reflexive, symmetric and transitive, then the relation is called an "equivalence relation" and denoted by in general.

**DEFINITION2:** Let be an equivalence relation over a set and . The set defined as is called the “equivalence class” of under and denoted by , or . Since for every , then . So the equivalence class is non-empty for every . The family of all the equivalence classes of the relation is called the “quotient set” of by and denoted by

I.e.,

.

**DEFINITION3:** Let be an equivalence relation over a set and . If , then is called a “representative class” of the equivalence class .

**PROPOSITION1:** Let be an equivalence relation over a set and . Then,

.

**THEOREM1:** Let be an equivalence relation over a set . Then the quotient set is a partition of .

**EXAMPLE1:** Define as . (Where is an arbitrary constant in ).

**(i)** is reflexive because .

**(ii)** is symmetric because

.

**(iii)** is transitive because

.

Consequently is an equivalence relation over . We write . Let’s examine the equivalence classes of this equivalence relation in case of : For this, we will separately find the equivalence classes of the numbers and . According to Theorem1, because . Let be an even integer. Since , then . Hence, . Now, let be an odd integer. Since , then . Hence, . If and denote the even integers and the odd integers respectively, then and . There is no another equivalence class of this equivalence relation because any integer is either even or odd. Hence, according to Theorem1, and . More generally, all the equivalence classes of the relation are . Consequently, according to Theorem1, for all distinct ,

and the following equality are true:

**EXAMPLE2:** Let be the set of all the lines lying on the plane. Since a line isn’t orthogonal to itself, the orthogonality isn’t an equivalence relation among all the lines.

**EXAMPLE3:** Let be the parallelism relation over the set of all the lines lying on the plane. I.e.,

**(i)** is reflexive because every line is parallel to itself.

**(ii)** is symmetric because .

**(iii)** is transitive because .

(Where are arbitrary lines on the plane)

Hence, the parallelism is an equivalence relation among all the lines.

**EXAMPLE4:** Let f be a function from to . Define as .

**(i)** is reflexive because .

**(ii)** is symmetric because .

**(iii)** is transitive because

.

Hence, is an equivalence relation over the domain of the function .

Theorem1 shows that all the equivalence classes of an equivalence relation over a non-empty set is a partition of this set. Now, we will show the opposite. I.e., we will prove that when it’s given a partition of a non-empty set, there is one and only one equivalence relation, the equivalence classes of which are the parts of the partition.

**THEOREM2:** Let be a partition of a set . (Where , and ) Define as . Then is an equivalence relation, the equivalence classes of which are the sets of the family .