Is a relation from a set $A$ to a set $B$ always a proper subset of $A\times B$?

Question

Is a relation from a set $A$ to a set $B$ always a proper subset of $A\times B$? Or, is it possible that the relation covers the entire set $A\times B$?

Answer 1

Recall that the definition of a relation $R$ between two sets $A$ and $B$ is a subset $R \subseteq A \times B$ so relations are not always a proper subset of $A \times B$.

For example, given $A = \{a,b,c\}$ and $B = \{d,e,f\}$, we have $R = \{(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)\}$ which covers the entire set $A \times B$.

Answer 2

It is entirely valid to have a relation that relates every element of $A$ to every element of $B$.

It can even have fancy properties -- for example if $B$ is a singleton set, the relation $A\times B$ will be a function!

Answer 3

If either $A$ or $B$ are empty, then $A\times B=\varnothing$. In that case there is only one relation between the two sets. The empty relation. And there are certainly no proper subsets to $A\times B$.