Question about the definition of the composition of relations

Question

I am slightly confused about following definition:

Suppose R is a relation from A to B and S is a relation from B to C. Then the composition of S and R is the relation S ◦ R from A to C defined as follows: $$S \circ R = \{(a,c) \in A \times C \mid \exists b \in B((a,b) \in R \text{ and } (b,c) \in S\}$$

I struggle to explain my question in words, hence I will give you an example to show where I'm confused.

Case 1

Suppose

$A = \{a_1,a_2,a_3\}$, $B = \{b_1,b_2,b_3\}$, $C = \{c_1,c_2,c_3\}$

$R$ is a relation from $A$ to $B$, and $D$ is a relation from $B$ to $C$

$R = \{(a_1,b_1)\}$ and $D = \{(b_1,c_3)\}$

I'm pretty sure that in this case, $R \circ D = \{(a_1,c_3)\}$

But now consider slightly different example:

Case 2

$A = \{a_1,a_2,a_3\}$, $B = \{b_1,b_2,b_3\}$, $C = \{c_1,c_2,c_3\}$

$R$ is a relation from $A$ to $B$, and $D$ is a relation from $B$ to $C$

$R = \{(a_1,b_2)\}$ and $D = \{(b_1,c_3)\}$ (Note that each subsets of $R$ and $D$ now have $b$'s with different indices)

In this case, is $R \circ D = \{(a_1,c_3)\}$ or $R \circ D = \emptyset$ ?

Answer 1

Note that your definition is understand by right to left:

First you take $(a, \color{red}b) \in R$ and if $( \color{red}b,c) \in S$, then add $(a,c)$ to $S \circ R$

So your computation in case 1 is wrong!, since $(\color{red}{c_3}, ?) \notin R $

For case 2, it is empty!

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Note that a relation from $A$ to $C$ is any subset of $A \times C$, so empty set is also a relation. See here

Answer 2

You are correct.

The composite relation is not always non-empty.

You have to have the range of $D$ to be a subset of the domain of $ R$ in order for $RoD$ to be non-empty.