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

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Please find below the solution to the asked query:

Let R be a relation which is an equivalence relation.Let x,yR-1. Then y,xR. Since R is symmetric, x,yR, thus y,xR-1,hence R-1 is a symmetric relation.Let x,yR-1. Then y,xR. Since R is symmetric, x,yR, then x,xR as it is reflexive, hence x,xR-1, hence R-1, is a reflexive relation.Let x,y and y,zR-1, hence y,x and z,yR, and as R is transitive, hencez,xR, hence x,zR-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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