For any relation R in any set A, we can define the inverse relation R^{-1} by a relation a R^{-1} b if and only if bRa.Prove that R is symmetric if and only if R=R^{-1}

(x,y)$\in $R

if and only if xRy

if and only if yRx (because R is symmetric)

if and only if (y,x)$\in $R

if and only if x R

^{-1}y

Hence R= R

^{-1}

Conversely suppose that R= R

^{-1}

Let xRy

(x,y)$\in $ R

(y,x)$\in $R

^{-1}

(y,x)$\in $R(because R= R

^{-1})

So, R is symmetric.

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