Multiset

The bags that can be used for shopping are at bag.

A multiset (sometimes called a bag) is a concept from mathematics. In many ways, multisets are like sets. Certain items are either elements of that multiset, or they are not. However, multisets are different from sets: The same type of item can be in the multiset more than once. For this reason, mathematicians have defined a relation (function) that tells, how many copies of a certain type of item there are in a certain multiset. They call this multiplicity. For example, in the multiset { a, a, b, b, b, c }, the multiplicities of the members a, b, and c are 2, 3, and 1, respectively. From a set of n elements, the number of r-element multisets is written as $\textstyle \left(\!{\binom {n}{r}}\!\right)$. This is sometimes called the multiset coefficient.

A multiset is illustrated by means of a histogram.

A multiset can also be considered an unordered tuple:

• The tuples (a,b) and (b,a) are not equal, and the tuples (a,a) and (a) are not equal either.
• The multisets {a,b} and {b,a} are equal, but the multisets {a,a} and {a} are not equal.
• The sets {a,b} and {b,a} are equal, and the sets {a,a} and {a} are equal too.

Examples

One of the simplest examples is the multiset of prime factors of a number n. Here, the underlying set of elements is the set of prime divisors of n. For example, the number 120 has the prime factorisation

$120=2^{3}3^{1}5^{1}$

which gives the multiset {2, 2, 2, 3, 5}.

Another is the multiset of solutions of an algebraic equation. A quadratic equation, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be { 3, 5 }, or it could be { 4, 4 }. In the latter case, it has a solution of multiplicity 2.

Related pages

• Combination

References

Sources

• Stanley, Richard P. (1997, 1999). Enumerative Combinatorics, Volumes 1 and 2. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1..