Give an example of relation $R$ and $S$ on $A$ such that $R$ and $S$ are nonempty, and $R \circ S$ and $S \circ R$ are empty

Question

Let $A = \left \{a, b, c, d\right \}$, give an example of relation $R$ and $S$ on $A$ such that $R$ and $S$ are nonempty, and $R \circ S$ and $S \circ R$ are empty

I'm thinking of ways that a set could be empty, and the only thing I can think of is that if two sets are disjoint, then their intersection is empty. However, I'm not sure as to how to go about coming up with the relations. Could someone give me a hint.

Answer 1

How about the simplest relations you can think of: $R=\{(a,a)\}$ and $S=\{(b,b)\}$

Why are $R\circ S$ and $S\circ R$ empty. Try to generalize.

Answer 2

The key to this problem is the definition of $R\circ S$. This is $\{(a,c):\exists b, (a,b)\in S, (b,c)\in R\}$. For $R\circ S$ to be empty, you need to make sure that the second components of all the pairs in $S$ are not among the first components of the pairs in $R$.