# [SOLVED] Composition of relations. Finding R^3

• Dec 8th 2009, 07:04 AM
sudeepmansh
[SOLVED] Composition of relations. Finding R^3
If A={1,2,3,4} and Relations R={(1,1),(1,2),(2,3),(3,4)} Find R^3?

Here R^3 means R o R o R.

First We calculate R o R. And Then we have to take the composition of RoR and R.

My question Whether we have to compute (RoR) o R or (RoR) o R , because both gives a different result.

The convention used here is, The composition of R and S is S o R.

R o R = { (1,2),(1,3),(2,4)}
(RoR) o R = {(1,2),(1,3),(1,4)}
and R o (R o R) = {(1,3),(1,4)}

so Which R^3 among the above 2 is correct?
• Dec 8th 2009, 08:28 AM
Plato
Quote:

Originally Posted by sudeepmansh
If A={1,2,3,4} and Relations R={(1,1),(1,2),(2,3),(3,4)} Find R^3?
My question Whether we have to compute (RoR) o R or (RoR) o R , because both gives a different result.
R o R = { (1,2),(1,3),(2,4)}
(RoR) o R = {(1,2),(1,3),(1,4)}
and R o (R o R) = {(1,3),(1,4)}

In fact they are not different.
You missed this: $R\circ R=\{{\color{blue}(1,1)},(1,2),(1,3),(2,4)\}
$

It is a theorem: If each of $R,~S,~\&~T$ is a relation on a set $\mathcal{A}$ then $R\circ (S\circ T)=(R\circ S)\circ T.$