What is the composition of the two given relations $R_1\circ R_2$?

Question

I have a set $A = \{a,b,c,d\}$ on which two relations are $R_1=\{(a,b),(a,d),(b,c),(c,a),(c,d),(d,b)\}$ and $R_2=\{(a,b),(b,c),(d,c),(a,d),(a,c)\}$. What will $R_1\circ R_2$ be? $\circ$ is a the compose after mapping. I don't understand.

Answer 1

Hint: A pair $(x,y)$ is in $R_1\circ R_2$ precisely if you can find a $z$ with $(x,z)$ in $R_2$ and $(z,y)$ in $R_1$ (there may be more than one such "enabling" $z$ for a given $(x,y)$, but that doesn't matter--there just has to be at least one).

For example, $(\boxed{a},\boxed{c})$ is in $R_1\circ R_2$ because $(\boxed{a},b)$ is in $R_2$ and $(b,\boxed{c})$ is in $R_1$. In this case, we found $b$ that did the trick.

Now you just need to examine the various cases.

You could go through the elements of $R_2$ one at a time and see which ones have a second coordinate that matches the first coordinate of anything in $R_1$. That will give you a pair in the composition (you can read the other coordinate of the pair off the match -- there may be more than one such pair).

For example, using $(a,c)$ from $R_2$, the $c$ matches the first coordinate of the pairs $(c,\boxed{a})$ and $(c,\boxed{\boxed{d}})$ in $R_1$. That means that both $(a,\boxed{a})$ and $(a,\boxed{\boxed{d}})$ are in the composition.

Now do the same thing one at a time with each of the other pairs in $R_2$.