Question

If the relation $R:A\to B$ where $A=\{1,2,3,4\}$, $B=\{1,3,5\}$ is defined by $R=(x,y):x<y,x\in A,y\in B$, then $RO{R}^{-1}$ is

• A. $\left\{\right(1,3,),(1,5,),(2,3,),(2,5),(3,5,),(4,5,\left)\right\}$
• B. $\left\{\right(3,1),(5,1),(5,2),(5,3,),(5,4\left)\right\}$
• C. $\left\{\right(3,3),(3,5),(5,3),(5,5\left)\right\}$
• D. $\left\{\right(3,5\left)\right\}$

Answer 1

The correct option is C $\left\{\right(3,3),(3,5),(5,3),(5,5\left)\right\}$

As given in the question

$R=\left\{\begin{array}{ccc}(1,3)& (1,5)& (2,3)\\ (2,5)& (3,5)& (4,5)\end{array}\right\}$

Therefore, $R^{-1}=\left\{\begin{array}{ccc}(3,1)& (5,1)& (3,2)\\ (5,2)& (5,3)& (5,4)\end{array}\right\}$

Now if $R$ and $S$ are two relations then $ROS$ has elements $(x,z)$ if $(x,y)\in S$ and $(y,z)\in R$.

Thus, for two relations $R$ and $R^{-1}$, applying above result we get $RO{R}^{-1}=\left\{\begin{array}{c}(3,3),(3,5),(5,3),(5,5)\end{array}\right\}$