Question

If $R$ be a relation $<$ from $A=\{1,2,3,4\}$ to $B=\{1,3,5\}$ i.e., $(a,b)\in$ R if $a<b$, then find $RO{R}^{-1}$

Answer 1

$A=\{1,2,3,4\}$
$B=\{1,3,5\}$
$R=\{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)\}$
$R^{-1}=\{(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)\}$

For composition $RO{R}^{-1}$, we will pick up an element of $R^{-1}$ first then of $R$.
eg: $(3,1)\in R^{-1}$ $(1,3)\in R$ $\Rightarrow (3,3)\in RO{R}^{-1}$
$\therefore RO{R}^{-1}=\{(3,3),(3,5),(5,3),(5,5)\}$