Axiom:Axiom of Choice/Formulation 3

Axiom

Let $\SS$ be a set of non-empty pairwise disjoint sets.

Then there exists a set $C$ such that for all $S \in \SS$, $C \cap S$ has exactly one element.

Symbolically:

$\forall s: \paren {\paren {\O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O} \implies \exists c: \forall t \in s: \exists x: t \cap c = \set x}$

That is, there exists a choice set $C$ for $\SS$.

Also known as

This form of the Axiom of Choice is sometimes called the Axiom of Selection.

Also see

• Definition:Transversal

• Equivalence of Formulations of Axiom of Choice

• Results about the Axiom of Choice can be found here.

Sources

• 1908: Ernst Zermelo: Neuer Beweis für die Möglichkeit einer Wohlordnung ("A new proof of the possibility of well-ordering") (Math. Ann. Vol. 65: pp. 107 – 128)
• 1973: Thomas J. Jech: The Axiom of Choice ... (previous) ... (next): $1.$ Introduction: $1.4$ Problems: $1$
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Exercise $1.5$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): axiom of choice