A bag, or multiset, is a set whose members are not necessarily distinct, i.e., it can contain duplicates. I can't think of a way to say this in a classical first-order language with equality, i.e., using only variables, function symbols, predicate symbols, implication (or conjunction or disjunction), negation, and universal (or existential) quantifier. Do I need to add modal operators? How do I say 'x shows up y many times in z'? What if I take bags as undefined and define sets in terms of bags? This shouldn't be stumping me, so I only want a hint please.(adsbygoogle = window.adsbygoogle || []).push({});

Oh, I suppose the 'power set' of a multiset could have some equal sets. So I can maybe define a set as a multiset whose power set contains no equal sets?

Multisets are also just unordered n-tuples.

If I keep the normal set definitions, noting that the members of my domain are sets and

[tex]\forall x \exists z \forall y (y \in z \leftrightarrow y \subseteq x)[/tex],

where z is called the power set of x, denoted P(x), then a multiset w is a set such that the cardinality of P(w) can be less than the cardinality of w.

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Okay, nevermind. I looked it up.

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# First-order, non-modal definition of 'bag/multiset'

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