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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.

Composition

1 Answers1

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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.

giobrach
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  • Your answer is really good, but I am sorry to say I still dont really get how the black arrows get there –  Dec 30 '17 at 17:58
  • Look at the definition of relation at the top of the picture. It is basically saying “there is a black arrow going from $a$ to $c$ (that is, $(a,c)\in S\circ R$ whenever there is an element $b$ such that there is a blue arrow going from $a$ to $b$ (that is, $(a,b)\in R$), and there is a red arrow going from $b$ to $c$ (that is, $(b,c)\in S$).” – giobrach Dec 30 '17 at 18:03
  • Also, welcome to Math.SE! – giobrach Dec 30 '17 at 18:06
  • Okay now I understand why there has to be black arrows, but how do I know which way it should point i.e. the direction of the black arrow? And thank you! –  Dec 30 '17 at 18:07
  • If the couple $(x,y)$ is a member of some relation, that means that in your diagram you should be drawing an arrow going from $x$ to $y$, that is, with its foot on $x$ and its tip on $y$. So the arrow should point toward the latter element of the ordered couple, which would be $y$. – giobrach Dec 30 '17 at 18:11