Multiplicity

set-theory combinatorics

Definition

Multiplicity

The multiplicity of an object $x$ in a finite collection $M=(x_i{)}_{i\in I}$ is the number of times $x$ occurs:

$$m_M(x):=\left|\{i\in I\mid x_i=x\}\right|.$$

In a multiset, multiplicity is part of the collection: $m_M(x)$ may be any non-negative integer. In an ordinary set, only membership matters, so every object has multiplicity $0$ or $1$.

Example

BANANA

In the letters of BANANA,

$$m(A)=3,m(N)=2,m(B)=1.$$