Editing Multiset (section)

==Basic properties and operations==
Elements of a multiset are generally taken in a fixed set {{mvar|U}}, sometimes called a ''universe'', which is often the set of [[natural number]]s. An element of {{mvar|U}} that does not belong to a given multiset is said to have a multiplicity 0 in this multiset. This extends the multiplicity function of the multiset to a function from {{mvar|U}} to the set <math>\N</math> of non-negative integers. This defines a [[one-to-one correspondence]] between these functions and the multisets that have their elements in {{mvar|U}}.

This extended multiplicity function is commonly called simply the '''multiplicity function''', and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the [[indicator function]] of a [[subset]], and shares some properties with it.

The '''support''' of a multiset <math>A</math> in a universe {{mvar|U}} is the underlying set of the multiset. Using the multiplicity function <math>m</math>, it is characterized as 
<math display="block">\operatorname{Supp}(A) := \{x \in U \mid m_{A}(x) > 0 \}.</math>

A multiset is ''finite'' if its support is finite, or, equivalently, if its cardinality
<math display="block">|A| = \sum_{x\in\operatorname{Supp}(A)} m_A(x) = \sum_{x\in U} m_A(x)</math>
is finite. The ''empty multiset'' is the unique multiset with an [[empty set|empty]] support (underlying set), and thus a cardinality 0.

The usual operations of sets may be extended to multisets by using the multiplicity function, in a similar way to using the indicator function for subsets. In the following, {{mvar|A}} and {{mvar|B}} are multisets in a given universe {{mvar|U}}, with multiplicity functions <math>m_A</math> and <math>m_B.</math>

* '''Inclusion:''' {{mvar|A}} is ''included'' in {{mvar|B}}, denoted {{math|''A'' ⊆ ''B''}}, if <math display="block">m_A(x) \le  m_B(x)\quad\forall x\in U.</math>
* '''Union:''' the ''union'' (called, in some contexts, the ''maximum'' or ''lowest common multiple'') of {{mvar|A}} and {{mvar|B}} is the multiset {{mvar|C}} with multiplicity function<ref name=syropoulos/> <math display="block">m_C(x) = \max(m_A(x),m_B(x))\quad\forall x\in U.</math>
* '''Intersection:''' the ''intersection'' (called, in some contexts, the ''infimum'' or ''greatest common divisor'') of {{mvar|A}} and {{mvar|B}} is the multiset {{mvar|C}} with multiplicity function <math display="block">m_C(x) = \min(m_A(x),m_B(x))\quad\forall x\in U.</math>
* '''Sum:''' the ''sum'' of {{mvar|A}} and {{mvar|B}} is the multiset {{mvar|C}} with multiplicity function <math display="block">m_C(x) = m_A(x) + m_B(x)\quad\forall x\in U.</math> It may be viewed as a generalization of the [[disjoint union]] of sets. It defines a [[commutative monoid]] structure on the finite multisets in a given universe. This monoid is a [[free commutative monoid]], with the universe as a basis.
* '''Difference:''' the ''difference'' of {{mvar|A}} and {{mvar|B}} is the multiset {{mvar|C}} with multiplicity function <math display="block">m_C(x) = \max(m_A(x) - m_B(x), 0)\quad\forall x\in U.</math>

Two multisets are ''disjoint'' if their supports are [[disjoint sets]]. This is equivalent to saying that their intersection is the empty multiset or that their sum equals their union.

There is an inclusion–exclusion principle for finite multisets (similar to [[inclusion–exclusion principle|the one for sets]]), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an [[parity (mathematics)|odd]] number of the given multisets, while in the second sum we consider all possible intersections of an [[parity (mathematics)|even]] number of the given multisets.{{Citation needed|date=August 2019}}