Definition:Empty Mapping

Theorem
For each set $T$ there is only one mapping for which the domain is the empty set:


 * $\varnothing \subseteq \varnothing \times T = \varnothing$

This is called:


 * The null mapping (or function);
 * The empty mapping (or function).

The null relation $\mathcal R = \varnothing \subseteq S \times T$ is not a mapping unless $S = \varnothing$.

Proof
Suppose $S \ne \varnothing$.

From the definition of an empty set, $S \ne \varnothing \implies \exists x \in S$.

Thus: