| 11. | This is why the above result only gives an equivalence between nonempty rectangular bands and pairs of nonempty sets.
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| 12. | The domain of discourse " D " is a nonempty set of " objects " of some kind.
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| 13. | Let \ rho _ X, \ rho _ Y be extended pseudometrics on nonempty sets X, Y, respectively.
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| 14. | States that there is a nonempty set which is closed under the predecessor and successor operations and yet does not contain all numbers.
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| 15. | The axiom of choice produces a choice set whose size is not bigger than the size of the given set of nonempty sets.
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| 16. | The complex plane cannot be entirely covered by " n " disjoint open nonempty sets for " n " > 1.
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| 17. | Here's the general proof : suppose " X " is a nonempty set that you wish to prove has a least element.
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| 18. | The only thing that is given is the family { a _ i } which consists of nonempty sets not non empty " disjoint " sets.
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| 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.
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| 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.
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