I have a very hard time understanding composition of relations. The following is a composition I am trying to understand, but I cannot figure out where the composite pairs and the power pairs comes from.
2026-04-15 12:38:56.1776256736
Relation composition
1.1k Views Asked by user504783 https://math.techqa.club/user/user504783/detail At
1
It might be useful to work with diagrams. Write $1$, $2$, and $3$ in the triangular disposition that you find in the picture. Now draw the arrows corresponding to the relation $R$ in blue. On the same diagram, draw the arrows corresponding to $S$ in red. The relation $S\circ R$ ($S$ after $R$) will correspond to new (black) arrows built in the following fashion.
Notice there is a blue arrow going from $1$ to $2$ and a red arrow going from $2$ to $3$. This means that there will be a black arrow going from $1$ to $3$. Repeat this procedure for all pairs of blue-red arrows such that the tip of the blue arrow coincides with the foot of the red arrow.
The relation $R \circ S$ is the same story, but you should follow red arrows before blue arrows on your diagram.
The relation $R \circ R$, also written $R^2$, works the same way: you should draw the arrows corresponding to $R$ first in blue and then in red (so the same arrows twice in the two different colors), and repeat the above procedure. Once you have $R^2$, higher powers are easy to obtain.