Axiom of Choice: Family of non-empty sets mutually disjoint or not?

I have noticed that the family of non-empty sets referred to in statements of the Axiom of Choice is sometimes required to be mutually disjoint, and sometimes not. Why is that?

Solution 1:

It makes no real difference: the two statements are equivalent. The version with pairwise disjoint sets is conceptually simpler, while the other is slightly easier to apply, since it's superficially more general.

Solution 2:

Let me add on Brian's answer.

If $A$ is a family of non-empty sets then there is a function $F\colon A\to\bigcup A$ such that $F(a)\in a$ for all $a\in A$.

This is the usual axiom of choice, we didn't require that $A$ is a pairwise disjoint family here.

If $A$ is a family of pairwise disjoint, non-empty sets then there is $C$ such that for all $a\in A$, $|a\cap C|=1$.

This statement is sometimes the given formulation of the axiom of choice, and here we have to require that $A$ is a pairwise disjoint family. Otherwise $\{\{0,1\},\{1,2\},\{0,2\}\}$ will have no such $C$.

The two are of course equivalent. But looking just at the assumption in the statement one requires disjointness, and one doesn't.