So I am not told if these statements are true or not (although they appear true). I am asked to either prove them in general terms, or disprove them with specific examples. I am confused about representing the unions below in 'general' terms, and then using composition. This is the question:
P,Q,R are binary relations on set S such that P,Q,R ⊆ S x S. Prove or disprove the following statements:
- (P ∪ Q);R = (P;R) ∪ (Q;R)
- P;(Q ∩ R) = (P;Q) ∩ (P;R)
this is a standard exercise on relations and sets. Basically, you want to show that both sides are (not) the same sets using only definitions.
For example, assume you have a pair $(a,b)\in(P\cup Q)\circ R$. That means, there is an element $c\in S$ such that $(a,c)\in R$ and $(c,b)\in P\cup Q$. The latter condition gives us, by definition of union, $(c,b)\in P$ or $(c,b)\in Q$. Now we need to combine the fact former $(a,c)\in R$ with the statement "$(c,b)\in P$ or $(c,b)\in Q$". Again, by definition of composition, we get that $(a,b)\in P\circ R$ or $(a,b)\in Q\circ R$. But this means $(a,b)\in(P\circ R)\cup(Q\circ R)$, therefore $(P\cup Q)\circ R\subseteq(P\circ R)\cup(Q\circ R)$. In the same fashion you have to show the other inclusion.
The second exercise is similar. Can you finish it?