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Equivalence Relation

In mathematics, an equivalence relation is denoted by R or $\sim$ or $\equiv$. A relation is said to be an equivalence relation if and only if it is -

  • Reflexive
  • Symmetric
  • Transitive
We can say that a relation is equivalence relation if it is reflexive, symmetric and transitive. Also, if a relation is reflexive, symmetric, transitive, then it is equivalence relation.
Let us study each in detail.

Reflexivity: A relation defined on a set is a reflexive relation if each element of it is mapped to itself. In other words, a relation R on a set S is said to be reflexive if
a R a, $\forall$ a $\epsilon$ S
Equality over a set real numbers is a reflexive relation.
For example: 10 = 10


Symmetry: A relation defined on a set is a reflexive relation if an element "a" is related to another element "b", then "b" must be related to "a". Therefore, a relation R on a set S is known as a symmetric relation -
if a R b $\Rightarrow$ b R a $\forall$ a, b $\epsilon$ S
a = b is a symmetric relation, since a = b $\Rightarrow$ b = a, where a and b are real numbers.
For example: If 2 = 3, then 3 = 2


Transitivity: Transitive relation on a set is defined as - if an element "a" is related to another element "b" and "b" is related to "c", then "a" must be related to "c". We can also write as -
if a R b and b R c $\Rightarrow$ a R c $\forall$ a, b, c $\epsilon$ S
Where, R is a symmetric relation defined over the set S. Equality is also a transitive relation.

Since the relation "is equal to" or equality satisfies all three necessary conditions of equivalence, hence it is an equivalence relation.

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