How to intuitively think about composition of relations?

Question

Examples:

1. $A \cap B$

   elements that are in both $A$ and $B$

2. $A \cup B$

   elements that are in either $A$ or $B$

3. $R \subset A\times B$

   coordinates where the first coordinate is an element of $A$, and the second is an element of $B$

4. $S \subset B\times C$

   coordinates where the first coordinate is an element of $B$, and the second is an element of $C$

5. $R^{-1}$

   coordinates of $R$, but the first and second coordinate are flipped

6. $S\circ R$

Is the final example something like this?: coordinates where the first coordinate is the first coordinate of a coordinate in $R$, and the second coordinate is the second coordinate in a coordinate in $S$, AND the second coordinate in $R$, and the first coordinate in $S$ are both in the same set

Answer 1

For intuition, you could think of $S \circ R$ as consisting of the ordered pairs that come from merging an ordered pair $(a, b) \in R \subset A \times B$ with an ordered pair $(b, c) \in S \subset B \times C$ to get an ordered pair $(a, c) \in A \times C$. (Note that the second coordinate of the pair from $R$ is the same as the first coordinate of the pair from $S$.)