Question

give an example of a choice function on the collection.

X = {{0.1},{2,3},{4,5}}

Answer 1

Axiom of Choice.

Let $X$ be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function $f$ defined on $X$ with the property that, for each set $S$ in the collection, $f(S)$ is a member of $S.$

The function $f$ is then called a choice function.

For example, if $X$ is the collection $X =\{ \{{0,1\}} ,\{{2,3\}},\{{4,5\}}\}$, then we can define $f$ quite easily: just let $f(S)$ be the smallest member of $S.$

Then the function that assigns 0 to the set $\{{0,1\}},$ 2 to $\{{2,3\}},$ and 4 to $\{{4,5\}}$ is a choice function on X.