Choice Function/Examples/Doubleton
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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/>.