Which one of the following relations on R is an equivalence relation?
Explanation for correct option:
Let be a relation
Reflexive relation: A homogeneous binary relation on a set X is reflexive if it relates every element of to itself.
Symmetric relation: a relation R is symmetric only if is true when .
Transitive relation: In transitive relation i.e., and then it implies
Equivalence relation: if a relation satisfies all the conditions of reflexive, symmetric, and transitive relation then we say that the relation has an equivalence relation.
For option (a)
For reflexive it is always true that if
For symmetric relation
let
This relation satisfies symmetric relation.
For transitive relation
let
This relation satisfies the transitive relation.
satisfies reflexive relation, symmetric relation, and transitive relation.
Hence this expression satisfies the equivalence relation.
Therefore, option (a) is the correct option.
Explanation for incorrect options:
Option (B)
If , then .
So does not satisfy the symmetric property.
Hence is not an equivalence relation.
Option (C)
does not imply .
So does not satisfy the symmetric property.
Hence is not an equivalence relation.
Option (D)
If , then it does not imply .
So does not satisfy the symmetric property.
Hence is not an equivalence relation.
Therefore option (A) is correct.