Exercise · ZW-AXIOM-OF-CHOICE-005 · ★☆☆ Basic
What does the axiom of choice guarantee for a family \(X\) of nonempty sets?
Choose one option.
Show answer
- C. That at least one choice function on \(X\) exists.
Explanation
The axiom guarantees a choice function exists, but says nothing about the size of \(X\) or its members. Confusing choice with cardinality is a common mistake.
Review the concept: Axiom of choice
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
- ZW-AXIOM-OF-CHOICE-001 What does the axiom of choice state?
- ZW-AXIOM-OF-CHOICE-002 Given a family X of nonempty sets, what must a choice function f satisfy?
- ZW-AXIOM-OF-CHOICE-003 Which of these statements is equivalent to the axiom of choice over ZF?
- ZW-AXIOM-OF-CHOICE-004 In what context is the axiom of choice formulated as an independent axiom?
- ZW-AXIOM-OF-CHOICE-006 Let X=\\1,2\,\3,4,5\\. Is the axiom of choice needed to find a choice function on X?
- ZW-AXIOM-OF-CHOICE-007 For an infinite family of pairs of indistinguishable socks, what is needed to choose one sock from…
- ZW-AXIOM-OF-CHOICE-008 Let f:A→ B be surjective. Using the axiom of choice, which function g:B→ A can be constructed?
- ZW-AXIOM-OF-CHOICE-009 A student claims that the family \(0,n): n∈ℕ, n≥ 1\ admits no choice function without the axiom. Wh…