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:

$$ $$ $$