Question

If R is a symmetric relation on a set A, then write a relation between R and ${{R}^{-1}}$.

Answer 1

Hint: For solving this problem, we first let a set A have two elements (x, y). Now, by using the definition of symmetric relation, we try to find out R and ${{R}^{-1}}$. After obtaining both the values, we can easily write the relation between them.

Complete step-by-step answer:
First, we assume a set A having two elements x and y such that $\left( x,y \right)\in A$. Now, the condition for a relation to be symmetric can be stated as: $\left( a,b \right)\in R\Rightarrow \left( b,a \right)\in R$
By using the above condition, we can write the symmetric relation for set A can be explained in two ways. First way of representation includes the consideration of x as the first element and the second way of representation is the consideration of y as the first element. So, it can be shown as:
$\begin{align}
  & {A}'=\left\{ \left( x,y \right),\left( y,x \right) \right\}\text{ for all x,y}\in A \\
 & {A}''=\left\{ \left( y,x \right),\left( x,y \right) \right\}\text{ for all x,y}\in A \\
\end{align}$
Now, the relation R can be stated as: $R=\left\{ \left( x,y \right),\left( y,x \right) \right\}\text{ for all x,y}\in A\ldots \left( 1 \right)$
Also, the relation ${{R}^{-1}}$ can be stated as: ${{R}^{-1}}=\left\{ \left( y,x \right),\left( x,y \right) \right\}\text{ for all x,y}\in A\ldots \left( 2 \right)$
From equation (1) and equation (2), we get $R={{R}^{-1}}$.
Therefore, both the relations are equal.

Note: The key steps involved in solving this problem is the knowledge of symmetric relation of a set. Also, the symmetry function enables the interchanging of variables within a parameter. So, the relationship between R and ${{R}^{-1}}$ must be equal.