I understand the null relation ($\emptyset \subset A \times B)$ is simply the relation that relates NO element of set $A$ to any element of set $B$.
But what about in the context of composition? For example, if a relation $R$ maps from $C$ to $A$ (i.e. $R \subset C \times A$), then what is $R \circ S $, if $S$ is the empty relation?
Would $R \circ S$ also be the empty relation? So $R \circ S = S$? Because $R$ relates no elements of $C$ to $B$?
Note that here I'm defining a composition of relations to first "apply" the outer relation, in this case $R$, then apply the inner relation, in this case the null/empty relation.
Empty sets are always fun.
Let's take an element $(x,y)$ of $R\circ\varnothing$.
This means that there exists an element $z$ such that $(x,z)\in R$ and $(z,y)\in\varnothing$.
...
$(z,y)\in\varnothing$...
...
that's absurd!
So the premise is false: we cannot take an element of $R\circ\varnothing$.
This means that $R\circ\varnothing=\varnothing$.