Question

If R $⊑$ A $\times$ B and S $⊑$ B $\times$ C be two relations, then $(SoR{)}^{-1}$ is equal to

• A. $S^{-1}o{R}^{-1}$
• B. RoS
• C. $R^{-1}o{S}^{-1}$
• D. None of these

Answer 1

The correct option is C $R^{-1}o{S}^{-1}$

SoR is relation from A to C.
$\therefore (SoR{)}^{-1}$ is a relation from C to A.
$R^{-1}$ is a relation from B to A.
$S^{-1}$ is a relation from C to B.
$\therefore R^{-1}0S^{-1}$ is a relation from C to A.
Let (C,a) $ϵ$ $(SoR{)}^{-1}$. $\therefore$ (a,c) $ϵ$ SoR
$\therefore$ $\exists$ b $ϵ$ B : (a,b) $ϵ$ R and (b,c) $ϵ$ s
$\therefore$ (b,a) $ϵ$ $R^{-1}$ and (c,b) $ϵ$ $S^{-1}$
$\therefore$ (c, a) $ϵ$ $R^{-1}$ $o{S}^{-1}$
$\therefore (SoR{)}^{-1}⊑R^{-1}o{S}^{-1}$
Conversely, Let (c, a) $ϵR^{-1}o{S}^{-1}$
$\therefore \exists$ b $ϵ$ B : (c, b) $ϵS^{-1}$ and (b,a) $ϵR^{-1}$
$\therefore$ (b,c) $ϵ$ S and (a,b) $ϵ$ R
$Rightarrow$ (a,c) $ϵ$ S and (a,b) $ϵ$ R
$\Rightarrow$ (a,c) $ϵ$ SoR $\Rightarrow$ (c,a) $ϵ$ $(SoR{)}^{-1}$
$\therefore R^{-1}o{S}^{-1}⊑(SoR{)}^{-1}$
Combining, we get $(SoR{)}^{-1}=R^{-1}o{S}^{-1}$