JEE Main & Advanced Mathematics Sets Composition of Relations

Let R and S be two relations from sets A to B and B to C respectively. Then we can define a relation SoR from A to C such that \[(a,\,\,c)\in SoR\Leftrightarrow \exists \,\,b\in B\] such that \[(a,\,\,b)\in R\] and \[(b,\,\,c)\in S\].

This relation is called the composition of R and S.

For example, if \[A=\text{ }\{1,\text{ }2,\text{ }3\},\,\,B=\text{ }\{a,b,c,d\},\,\,C=\{p,q,r,s\}\] be three sets such that \[R=\{(1,\,\,a),\,\,(2,\,\,b),\,\,(1,\,\,c),\,\,(2,\,\,d)\}\] is a relation from A to B and \[S=\{(a,\,\,s),\,\,(b,\,\,r),\,\,(c,\,\,r)\}\] is a relation from B to C. Then SoR is a relation from A to C given by \[SoR=\{(1,\,\,s)\,(2,\,\,r)\,\,(1,\,\,r)\}\]

In this case RoS does not exist.

In general \[RoS\ne SoR\]. Also \[{{(SoR)}^{-1}}={{R}^{-1}}o{{S}^{-1}}\].