Prove that the inverse of an equivalence relation is an equivalence relation - Maths - Relations and Functions

Question

Prove that the inverse of an equivalence relation is an equivalence
relation

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

Let R be a relation which is an equivalence relation
Let x,y∈R-1  Then y,x∈R
 Since R is symmetric, x,y∈R, thus y,x∈R-1,hence R-1 is a symmetric relation
Let x,y∈R-1  Then y,x∈R
 Since R is symmetric, x,y∈R, then x,x∈R as it is reflexive, hence x,x∈R-1, hence R-1, is a reflexive relation
Let x,y and y,z∈R-1, hence y,x and z,y∈R, and as R is transitive, hencez,x∈R, hence x,z∈R-1, hence  R-1, is a transitive relation
As R-1 is reflexive, symmetric and transitive, hence it is an equivalence relation
Hence inverse of an equivalence relation is an equivalence relation

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Prove that the inverse of an equivalence relation is an equivalence relation.