Question

Let a relation $R$ be defined by $R=\left\{\right(4,5),(1,4),(4,6),(7,6),(3,7\left)\right\}$. The relation $R^{-1}\circ R$ is given by

• A. $\left\{\right(1,1),(4,4),(7,4),(4,7),(7,7\left)\right\}$
• B. $\left\{\right(1,1),(4,4),(4,7),(7,4),(7,7),(3,3\left)\right\}$
• C. $\left\{\right(1,5),(1,6),(3,6\left)\right\}$
• D. None of these

Answer 1

Answer: (B)

$\left\{\right(1,1),(4,4),(4,7),(7,4),(7,7),(3,3\left)\right\}$

We have $R=\left\{\right(4,5),(1,4),(4,6),(7,6),(3,7\left)\right\}$.

$\therefore R^{-1}=\left\{\right(5,4),(4,1),(6,4),(6,7),(7,3\left)\right\}$

$(4,4)\in R^{-1}\circ R$ because $(4,5)\in R$ and $(5,4)\in R^{-1}$

$(1,1)\in R^{-1}\circ R$ because $(1,4)\in R$ and $(4,1)\in R^{-1}$

$(4,4)\in R^{-1}\circ R$ because $(4,6)\in R$ and $(6,4)\in R^{-1}$

$(4,7)\in R^{-1}\circ R$ because $(4,6)\in R$ and $(6,7)\in R^{-1}$

$(7,4)\in R^{-1}\circ R$ because $(7,6)\in R$ and $(6,4)\in R^{-1}$

$(7,7)\in R^{-1}\circ R$ because $(7,6)\in R$ and $(6,7)\in R^{-1}$

$(3,3)\in R^{-1}\circ R$ because $(3,7)\in R$ and $(7,3)\in R^{-1}$

$\therefore R^{-1}\circ R=\left\{\right(4,4),(1,1),(4,7),(7,4),(7,7),(3,3\left)\right\}$.

$\therefore$ The correct answer is $B$.