# Multiset

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A multiset is an 2-tulple (*S, f*), where [math]\displaystyle{ S }[/math] is a set and [math]\displaystyle{ f }[/math] is a frequency function (that returns the Frequency of a multiset member).

**AKA:**Bag, Unordered Sequence.**Context:**- It can be represented by a Frequency Table, or a Multiset Data Structure.
- It can be transformed by a Multiset Operation.
- It can range from being an Unordered Multiset to being an Ordered Multiset (e.g. a Tuple such as a Vector).
- It can range from being an Infinite Multiset to being a Finite Multiset (e.g. a Tuple of the marks reported for an course exam.).

**Example(s):**- M={a}.
- M={a, a}.
- M={1.1, 0.9, 1.1, 2.4}, a record of Experimental Measurement.
- {a, b, a} = {b, a, a}, a Multiset Operation.
- {a, b, a, a} ≠ {b, a, a}, a Multiset Operation.
- A Bag of Words.
- {A/6, B/11, C+/20, ..., F/2}, a set of marks reported for an course exam.
- {(a,1), (b,2), (a,1)}, a Tuple Multiset.
- A set of roots of a polynomial.
- a set of Sensor Values.
- a set of Random Experiments.
- …

**Counter-Example(s):**- (a, b, a) ≠ (b, a, a), a Tuple Operation.

**See:**Ordered Set, Set (Mathematics).

## References

### 2014

- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/multiset Retrieved:2014-4-21.
- In mathematics, the notion of
**multiset**(or bag) is a generalization of the notion of set in which members are allowed to appear more than once. For example, there is a unique set that contains the elements*a*and*b*and no others, but there are many multisets with this property, such as the multiset that contains two copies of*a*and one of*b*or the multiset that contains three copies of both*a*and*b*. The term "multiset" was coined by Nicolaas Govert de Bruijn in the 1970s.^{[1]}The use of multisets in mathematics and beyond predates the name "multiset" by many centuries: Knuth (1998) attributes the first study of multisets to the Indian mathematician Bhascara Acharya (circa 1150), who described permutations of multisets.

- In mathematics, the notion of

- ↑ Knuth also lists other names that were proposed for multisets, such as list, bunch, bag, heap, sample, weighted set, collection, and suite.