First slide

Relations XII

Question

If R $\subseteq$ A $\times$ B and A $\subseteq$ B $\times$ C be two relations, then (S o R)^-1 is equal to

Moderate

A

S^-1o R^-1
B

R o S
C

R^-1 o S^-1
D

None of these

Solution

S o R is relation from A to C
$∴$ (S o R)^-1is a relation from C to A.
R^-1is a relation from B to A
S^-1 is a relation from C to B.
$∴$ R^-1oS^-1is a relation from C to A.
Let (c, a) $\in$ SoR
$∴$ $\exists$ b $\in$ B :(a, b) $\in$ R and ( b, c) $\in$ S
$∴$ (b, a) R ^-1and (c, b) $\in$ S^-1
$∴$ ( c, a) $\in$ R^-1oS^-1
$∴$ (S o R)^-1 $\subseteq$ R^-1oS^-1
Conversely, let (c, a) $\in$ R^-1oS^-1
$∴$ $\exists$ b B: (c, b) $\in$ S ^-1 and (b, a) $\in$ R^-1
$∴$ (b, c) $\in$ S and (a, b) $\in$ R
$\Rightarrow$ (b, c) $\in$ S o R $\Rightarrow$ (c, a) $\in$ SoR^-1
$∴$ R^-1oS^-1 $\subseteq$ (S o R)^-1
combining , we get
(S o R)^-1= R^-1oS^-1

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