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.