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.

__

Okay, nevermind. I looked it up.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**