11. This is why the above result only gives an equivalence between nonempty rectangular bands and pairs of nonempty sets . 12. The domain of discourse " D " is a nonempty set of " objects " of some kind. 13. Let \ rho _ X, \ rho _ Y be extended pseudometrics on nonempty sets X, Y, respectively. 14. States that there is a nonempty set which is closed under the predecessor and successor operations and yet does not contain all numbers. 15. The axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets . 16. The complex plane cannot be entirely covered by " n " disjoint open nonempty sets for " n " > 1. 17. Here's the general proof : suppose " X " is a nonempty set that you wish to prove has a least element. 18. The only thing that is given is the family { a _ i } which consists of nonempty sets not non empty " disjoint " sets. 19. It states that given a collection of nonempty sets there is a single set " C " that contains exactly one element from each set in the collection. 20. However, some countably infinite sets of nonempty sets can be proven to have a choice function in ZF without " any " form of the axiom of choice.
