Difficulty understanding Axiom of Choice

Question

The Axiom of Choice is defined by Mcdonald, A Course in Real Analysis as follows:

"Suppose that $C$ is a collection of non-empty sets. Then there exists a function $f: C \rightarrow \bigcup\limits_{A\in C}A$ such that $f(A)\in A$ for each $A \in C$."

I don't understand what it means for a function to exist between a collection and a union of sets in the collection. Also, how do we know that putting $A$ into the function returns an element of $A$?

Answer 1

Perhaps an example will help.

Let $C = \{\{1,2\},\{2,3\},\{3,4\}\}$.

Notice that $\bigcup_{A \in C} A = \{1,2\} \cup \{2,3\} \cup \{3,4\} = \{1,2,3,4\}$.

The axiom of choice asserts, for this example, that

There exists a function $f : C \to \bigcup_{A \in C} A$ such that $f(A) \in A$ for each $A \in C$.

And this statement is true, as you can easily verify on your own without even applying the axiom of choice, by simply choosing correct values of the function $f$, which I will leave it for you to fill in: $$\begin{align*} f(\{1,2\}) &= ? \quad\text{(choose some element of the set $\{1,2\}$)} \\ f(\{2,3\}) &= ? \quad\text{(choose some element of the set $\{2,3\}$)}\\ f(\{3,4\}) &= ? \quad\text{(choose some element of the set $\{3,4\}$)} \end{align*} $$

The hard part comes when the set $C$ is infinite. In this case, you cannot always write down the infinitely many choices to be made in order to specify the choice function $f$. The Axiom of Choice is nonetheless verifying that this function exists, despite our inability to write it down.

Answer 2

The axiom says that from a collection of sets, you can choose one element in every set, by means of function.

So the domain of the function is the collection, and the range is the set of elements contained in the members of the collection.