## Econ: rational preference relation generates nonempty choice rule

Question: show that if $X$ is finite, then any rational preference relation generates a nonempty choice rule; that is, $C(B) \neq \emptyset$ for any $B \subset X$ with $B \neq \emptyset$

We know that if a preference relation is rational, then it is both transitive and complete. Then the choice structured generated satisfies the weak axiom. If we have all subsets of $X$ up to 3 elements as well, then then $C(B) = C*(B, >)$ for all $B$. Then it's just a matter of showing that $C*(B,>)$ is nonempty, which will in turn show that $C(B)$ is nonempty. However, we don't know if we have all subsets of $X$ up to 3 elements or not, so the proof cannot be done this way.

Any tips? Thanks!