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Equivalence Relation

Prove that if...

Question

Prove that if $R$ is an equivalence relation then $R^{- 1}$ is also.

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Solution

R be an equilence relation
$a R^{- 1} b = b R a$
R is an equilance relation
$a R a a R^{- 1} a R^{- 1}$ is reflexive
$\begin{matrix} a R^{- 1} b b R a a R b \end{matrix}$ systematic
$a R^{- 1} b b R^{- 1} c$
R is transitive
$c R a a R^{- 1} c$
$R^{- 1}$ is transitive
combining we say $R^{- 1}$ is equivalence

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