Multisets

Multisets (or bags) are unordered, nonuniqued collections. In this course, we won’t spend too much time with these sorts of objects, but it’s useful to know the terminology.

Multisets are often (somewhat confusingly) represented using the same notation as sets. For instance, the following is a multiset containing only vowels.

\[\bar{V}_1 \equiv \{\text{e, i, o, u, æ, ɑ, ɔ, ə, ɛ, ɪ, ʊ}\}\]

And this is a representation of the same multiset, since multisets are unordered.

\[\bar{V}_2 \equiv \{\text{o, u, æ, ɑ, ɔ, ə, ɛ, ɪ, ʊ, e, i}\}\]

But this is not a representation of a multiset, since multisets are nonuniqued.

\[\bar{V}_3 \equiv \{\text{o, o, o, u, u, æ, ɑ, ɔ, ə, ə, ə, ɛ, ɪ, ʊ, e, i}\}\]

Another way of saying this is that the multiplicity of a particular element matters in a multiset in a way it doesn’t matter in a set.

We often work with multisets in Python using dicts or specialized subclasses thereof. One special subclass of dict that is useful for representing multisets (and that you should know) is collections.Counter.

The nice thing about Counter is that it can be initialized with an iterable or mapping (such as a set) containing hashable objects (such as strs) and it will make a dictionary mapping the elements of that iterable/mapping to their multiplicity–i.e. how many times they show up in that iterable/mapping.¹

from pprint import pprint
from collections import Counter

vowels_bar_1: Counter[str] = Counter(
    ["e", "i", "o", "u", "æ", "ɑ", "ɔ", "ə", "ɛ", "ɪ", "ʊ"]
)

pprint(vowels_bar_1)

Counter({'e': 1,
         'i': 1,
         'o': 1,
         'u': 1,
         'æ': 1,
         'ɑ': 1,
         'ɔ': 1,
         'ə': 1,
         'ɛ': 1,
         'ɪ': 1,
         'ʊ': 1})

vowels_bar_2: Counter[str] = Counter(
    ["o", "u", "æ", "ɑ", "ɔ", "ə", "ɛ", "ɪ", "ʊ", "e", "i"]
)

pprint(vowels_bar_2)

Counter({'o': 1,
         'u': 1,
         'æ': 1,
         'ɑ': 1,
         'ɔ': 1,
         'ə': 1,
         'ɛ': 1,
         'ɪ': 1,
         'ʊ': 1,
         'e': 1,
         'i': 1})

vowels_bar_3: Counter[str] = Counter(
    ["o", "o", "o", "u", "u", "æ", "ɑ", "ɔ", "ə", "ə", "ə", "ɛ", "ɪ", "ʊ", "e", "i"]
)

pprint(vowels_bar_3)

Counter({'o': 3,
         'ə': 3,
         'u': 2,
         'æ': 1,
         'ɑ': 1,
         'ɔ': 1,
         'ɛ': 1,
         'ɪ': 1,
         'ʊ': 1,
         'e': 1,
         'i': 1})

And Counters behave as we would expect of a multiset–at least in terms of equality.

if vowels_bar_1 == vowels_bar_2:
    print("Vbar_1 = Vbar_2")
else:
    print("Vbar_1 ≠ Vbar_2")
    
if vowels_bar_1 == vowels_bar_3:
    print("Vbar_1 = Vbar_3")
else:
    print("Vbar_1 ≠ Vbar_3")

Vbar_1 = Vbar_2
Vbar_1 ≠ Vbar_3

Footnotes

1. Note that I’m passing the vowels in the multiset as a list to Counter. We crucially don’t want to pass them as a set, because that would destroy the multiplicities of the items.