Exercise · ZW-AXIOM-OF-CHOICE-006 · ★★☆ Intermediate
Let \(X=\{\{1,2\},\{3,4,5\}\}\). Is the axiom of choice needed to find a choice function on \(X\)?
Choose one option.
A Yes, because \(X\) contains sets with more than one element. B No, because \(X\) is finite and the choice can be made explicitly. C Yes, because the sets in \(X\) are not equipotent. D No, because \(X\) has no infinite sets, so no choice function exists. Show answer B. No, because \(X\) is finite and the choice can be made explicitly.← Back to the question
Explanation Since \(X\) is finite, one can pick an element from each set explicitly without invoking the axiom. Having several elements per set does not force its use.
Statement that every family of nonempty sets admits a choice function \(f\) that picks one element from each set in the family.
\[\forall X\,(\emptyset\notin X \rightarrow \exists f\,\forall A\in X\,(f(A)\in A))\]
More practice problems: axiom of choice