Null Relation is Mapping iff Domain is Empty Set

Theorem

Let $S$ and $T$ be sets.

The null relation $\RR = \O \subseteq S \times T$ is a mapping if and only if $S = \O$.

Proof

Sufficient Condition

Let $S = \O$.

Then the null relation $\RR = \O \subseteq S \times T$ is a mapping from Empty Mapping is Mapping.

$\Box$

Necessary Condition

Suppose $S \ne \O$.

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

Thus:

\(\ds \RR\)

\(=\)

\(\ds \O\)

\(\ds \leadsto \ \ \)

\(\ds \exists x \in S: \neg \exists y \in T: \, \)

\(\ds \tuple {x, y}\)

\(\in\)

\(\ds \RR\)

Definition of Empty Set

\(\ds \leadsto \ \ \)

\(\ds \RR\)

\(\)

\(\ds \text{ is not a mapping}\)

Definition of Mapping

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions