# 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

Let us assume that R is symmetric. Then if xRy then yRx.
(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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