Question

If $R$ be a relation from $A=\{1,2,3,4\}$ to $B=\{1,3,5\}$ that is $(a,b)\in R$ if $a<b$, then $R\circ R^{-1}$ is

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

Answer 1

The correct option is C

$\{(3,3),(3,5),(5,3),(5,5)\}$

Finding $R\circ R^{-1}$:

Given set $A=\{1,2,3,4\}$ and set $B=\{1,3,5\}$, then the relation $(a,b)\in R$ is obtained as the pair of each element from both the sets

Since in the relation $a<b$, then

$R=\{(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)\}$ then $R^{-1}$ is given by

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

To compute $R\circ R^{-1}$ pick one element from $R^{-1}$ and its correspondence relation $R$,

Take $(3,1)\in R^{-1}$ and $\left(1,3\right)\in R$ which implies $\left(3,3\right)\in R\circ R^{-1}$

Similarly, $(3,1)\in R^{-1}$ and $\left(1,5\right)\in R$ then $\left(3,5\right)\in R\circ R^{-1}$

Apply for other relations we obtain,

$R\circ R^{-1}=\{(3,3),(3,5),(5,3),(5,5)\}$

Hence, the correct option is (C).