Choice Function/Examples/Doubletons of Real Numbers

Example of Choice Function

Let $\FF$ be a set of sets of the form $\set {a, b}$ where $a$ and $b$ are real numbers.

Then there exists a choice function on $\FF$.

Proof

Let $f: \FF \to \bigcup \FF$ be the mapping defined as:

$\forall \set {a, b} \in \FF: \map f {\set {a, b} } = \map \min {a, b}$

where $\min$ denotes the minimum operation.

Then $f$ is a choice function on $\FF$.

$\blacksquare$

Sources

• 1973: Thomas J. Jech: The Axiom of Choice ... (previous) ... (next): $1.1$ The Axiom of Choice