sets theory-Realation proof question

Question

Let $A,B,C$ and $R,S,T$ be sets. And assume that

$$ R \subseteq A \times B, ~~ S \subseteq B \times C, ~~ T \subseteq B \times C.$$

Then, I want to show that $$ \begin{equation} (S \circ R) \cap (T \circ R) \subseteq (S\cap T) \circ R, \\ (S \circ R) \cup (T \circ R) = (S \cup T) \circ R. \end{equation} $$

I tried solving this for hours. I have no idea to even handle this. I would love any kind of help or assistence. Thank you.

Answer 1

If a(R o (S $\cap$ T))c, then exists
b in B with aRb and b(S $\cap$ T)c.
Thus aRb, bSc and bTc; a(RoS)c, a(RoT)c.
In conclusion, R o (S $\cap$ T) subset RoS $\cap$ RoT.

a(R o (S $\cup$ T))c iff exists b in B with
aRb, b(S $\cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $\cup$ T) = RoS $\cup$ RoT.

Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }