Is the empty set a relation?

Question

Is the empty set is a relation?

In Enderton's book A Mathematical Introduction to Logic, a relation is defined as a set of ordered pairs. If the empty set is a relation, why is that? In the text, there is an example of a function $\varnothing \to A$. This function is of course is the empty set, so it seems that the empty set is a relation. But I don't see the reason for this.

Answer 1

All the elements of the empty set are ordered pairs. To contradict this statement you will have to provide an element which is a counterexample, an element of the empty set which is not an ordered pair.

Since there is no such element, it follows that all the elements of the empty set are ordered pairs. Therefore the empty set is a relation.

Answer 2

Yes. Every element of the empty set is an ordered pair (vacuously), so the empty set is a set of ordered pairs.

Answer 3

It's probably best to say that a relation is an ordered triple $(A, B, R)$, where $R \subset A \times B$, so that the two sets on which the relation "operates" are explicit. With that convention, you can say that for any two sets $A$ and $B$, the triple $$ (A, B, \emptyset) $$ is a relation on $A$ and $B$.

The downside of using an ordered triple is that you then have to be careful about subsequent definitions where you might only want the "rule" part ($R$) of the relation...but that's not a big deal.

Answer 4

Consider the binary relation $R\subset A\times B$

Then $R=\emptyset \iff \forall x\in A: \forall y\in B: \neg x R y$

Likewise, $R=A\times B \iff \forall x\in A: \forall y\in B: x R y$

This works even if $A\times B=\emptyset$ (vacuously).