Choice Function/Examples/Doubleton

Examples of Choice Functions

Let $S$ be a doubleton.

Then there exist $2$ choice functions for $S$.

Proof

Let $S = \set {1, 2}$.

The non-empty subsets of $S$ are $\set 1$, $\set 2$ and $\set {1, 2}$.

A choice function for $S$ must include $1$ element from each of these subsets.

There is $1$ way to select $1$ element from a singleton.

There are $2$ ways to select $1$ element from a soubleton.

Hence by the Product Rule for Counting, there are $1 \times 1 \times 2$ choice functions for $S$.

These choice functions can be defined as:

$\map {f_1} {\set 1} = 1, \map {f_1} {\set 2} = 2, \map {f_1} {\set {1, 2} } = 1$

$\map {f_2} {\set 1} = 1, \map {f_2} {\set 2} = 2, \map {f_2} {\set {1, 2} } = 2$

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): choice (axiom of)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): choice, axiom of

• Bell, John L., "The Axiom of Choice", The Stanford Encyclopedia of Philosophy (Winter 2021 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2021/entries/axiom-choice/>.