Exercise · ZW-AXIOM-OF-CHOICE-007 · ★★☆ Intermediate
For an infinite family of pairs of indistinguishable socks, what is needed to choose one sock from each pair?
Choose one option.
A The axiom of choice, because no definable rule tells one sock from the other. B No choice at all, because each pair has two elements and you can take the first. C Nothing, because every infinite set admits a choice function without assumptions. D Only the principle of mathematical induction. Show answer A. The axiom of choice, because no definable rule tells one sock from the other.← Back to the question
Explanation Without a definable rule to tell the socks apart, choosing one from each of infinitely many pairs requires the axiom. Taking 'the first' assumes an order that does not exist here.
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