I'm reading Set Theory by Thomas Jech but can't figure out the reason of the statement below:
In some trivial cases, the existence of a choice function can be proved outright in ZF:
when every $X\in S$ is a singleton $X=\{x\}$.
But one cannot prove existence of a choice function (in ZF) just:
when every $X\in S$ is a set of two elements.
I can't figure out why. I'll appreciate it if you could prove the first statement and provide a counterexample of the second statement to let me understand them.