Multiset - Multiplicity Function

Multiplicity Function

The set indicator function of a normal set is generalized to the multiplicity function for multisets. The set indicator function of a subset A of a set X is the function

defined by

\mathbf{1}_A(x) = \begin{cases} 1 &\text{if }x \in A, \\ 0 &\text{if }x \notin A. \end{cases}

The set indicator function of the intersection of sets is the minimum function of the indicator functions

The set indicator function of the union of sets is the maximum function of the indicator functions

The set indicator function of a subset is smaller than or equal to that of the superset

The set indicator function of a cartesian product is the product of the indicator functions of the cartesian factors

The cardinality of a (finite) set is the sum of the indicator function values

Now generalize the concept of set indicator function by releasing the constraint that the values are 0 and 1 only and allow the values 2, 3, 4 and so on. The resulting function is called a multiplicity function and such a function defines a multiset. The concepts of intersection, union, subset, cartesian product and cardinality of multisets are defined by the above formulas.

The multiplicity function of a multiset sum, is the sum of the multiplicity functions

The multiplicity function of a multiset difference is the zero-truncated subtraction of the multiplicity functions

The scalar multiplication of a multiset by a natural number n may be defined as:

A small finite multiset, A, is represented by a list where each element, x, occurs as many times as the multiplicity, 1A(x), indicates.

Read more about this topic:  Multiset

Famous quotes containing the words multiplicity and/or function:

    “The point of cities is multiplicity of choice.”
    —Jane Jacobs (b. 1916)

    “If the children and youth of a nation are afforded opportunity to develop their capacities to the fullest, if they are given the knowledge to understand the world and the wisdom to change it, then the prospects for the future are bright. In contrast, a society which neglects its children, however well it may function in other respects, risks eventual disorganization and demise.”
    —Urie Bronfenbrenner (b. 1917)