We were given a list of properties for compositions of relations and asked to prove one of them as practice. I have tried different ways, including trying to get the cardinality of either side and looked through the notes but I don't understand where to start and how to actually prove this statement.
R ◦ (S1 ∩ S2) ⊆ (R ◦ S1) ∩ (R ◦ S2).
If anyone could point me in the right direction I'd greatly appreciate it, thanks.
On
To show that $\ A\subseteq B\ $ for any two sets $\ A\ $ and $\ B\ $ , you have to demonstrate that whenever $\ a\in A\ $, then $\ a\in B\ $.
If $\ T=S1\cap S2\ $, and you take an arbitrary pair $\ (x,y)\in R\circ T\ $, this means that there exists a $\ z\ $ such that $\ (x,z)\in R\ $ and $\ (z,y)\in T\ $. Can you now see how the pair $\ (z,y)\ $ must be related to $\ S1\ $ and $\ S2\ $?
HINT 1: Prove that $\in \circ(_1 \cap _2)\implies \in \circ _1 $ and $\in \circ _2$
HINT 2: Use the fact that $\in \circ \implies = R(y)$ for some $y\in S$.