Let $R = \{(1, 5), (2, 2), (3, 4), (5, 2)\}$, $S = \{(2, 4), (3, 4), (3, 1), (5, 5)\}$, and $T = \{(1, 4), (3, 5), (4, 1)\}$. Find
(1)$\quad R ∘ S$
(2)$\quad T ∘T.$
(3) $\quad T∘S$
Now from what I understand from the definition of composition
Consider that the $P$ is the relation from $A$ to $B$ and $Q$ is a relation from $B$ to $C$. The composite $P,Q$.
$$Q ∘ P = \{(a,c) : \text{there exists } b \in B \text{ such that } (a,b) \in P \text{ and } (b,c) \in Q\}$$
What I do not understand from this definition, is that is choosing $(x,y)$ arbitrary? As long as the conditions are satisfied? If that is the case can one assume that there are multiple answers for those questions.
For number one can the answer be
(1)$\quad R ∘ S = \{(2,3),(3,5)\}$
I just do not see the connection. If anyone could clear up these ambiguities that would be good.
There is no ambiguity in the definition of the composite relation. An element $a$ is related to an element $b$ through $R\circ S$ precisely if you can relate $a$ and $b$ through an auxiliary element $c$ — that is, a $c$ such that $a$ is $S$-related to $c$ and $c$ is $R$-related to $b$. It does not matter which element $c$ you use or even if there is more than one.
Let's do $(1)$. Look at the first element in $S$, namely $(2,4)$. So $2$ is $S$-related to $4$. Can $4$ serve as an auxiliary element to relate $2$ to something else? Unfortunately not, because $4$ is not $R$-related to anything. The same thing happens for $(3,4)$.
Now consider the third element, namely $(3,1)$. We are in luck, because $1$ is $R$-related to $5$. Therefore $3$ is $(R \circ S)$-related to $5$ through the auxiliary element $1$. Finally, the last element in $S$ is $(5,5)$. And again we are in luck, because $5$ is $R$-related to $2$. Therefore $5$ is $(R\circ S)$-related to $2$ through the auxiliary element $5$. We conclude that $R \circ S = \{ (3,5),(5,2) \}$.