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The total number of elements in a multiset, including repeated memberships, is the Cardinality of the multiset, and the number of times an element belongs to the multiset is the Multiplicity of that member. i.e. in the multiset {''a'', ''a'', ''b'', ''b'', ''b'', ''c''} the multiplicities of the members ''a'', ''b'', and ''c'' are respectively 2, 3, and 1, and the cardinality of the multiset is 6.

In multisets, as in sets, the order of elements does not matter; in contrast to Tuples , where order is important. The following list displays the difference between the three concepts, note how a multiset can also be considered an unordered tuple:

  • The tuples (''a'', ''b'') and (''b'', ''a'') are not equal, because in tuples order is important; and the tuples (''a'', ''a'') and (''a'') are not equal either, because in tuples and multisets Multiplicity is considered, affecting cardinality.


  • The multisets {''a'', ''b''} and {''b'', ''a''} are equal, because in multisets order is unimportant; but the multisets {''a'', ''a''} and {''a''} are not equal, as they have different cardinalities.


  • The sets {''a'', ''b''} and {''b'', ''a''} are equal, like the multisets {''a'', ''b''} and {''b'', ''a''}; but the sets {''a'', ''a''} and {''a''} are equal, unlike the multisets {''a'', ''a''} and {''a''}, this is because in sets, the concept of multiplicity does not exist, or in other words all objects, no matter how many times they belong to the set, still count as one.



FORMAL DEFINITION


Within from ''A'' to the set N = {1, 2, 3, ...} of ( Positive ) Natural Number s. The set ''A'' is called the ''underlying set of elements''. For each ''a'' in ''A'' the ''multiplicity'' (that is, number of occurrences) of ''a'' is the number ''m''(''a'').

The concept of a multiset is a Generalization of the concept of a set. A multiset is a set if the multiplicity of every element is one.

The function ''m'' is a set of Ordered Pairs { (''a'', ''m''(''a'')) : ''a'' in ''A'' }. For example, the multiset written as { ''a'', ''a'', ''b'' } is defined as { (''a'', 2), (''b'', 1) }, and the multiset { ''a'', ''b'' } is defined as { (''a'', 1), (''b'', 1) }.

An Indexed Family , ( ''ai'' ), where ''i'' is in some index-set, defines the multiset { ''ai'' } , provided no element occurs more than a finite number of times in the family. Even in an infinite multiset, the multiplicities must be finite numbers.


MULTIPLICITY FUNCTION

The set Indicator Function of a subset \ A of a set \ X is the function

:\mathbf{1}_A : X o \lbrace 0,1 brace \,

defined by

:\mathbf{1}_A(x) =
\left\{\begin{matrix}
1 &\mbox{if}\ x \in A, \
0 &\mbox{if}\ x
otin A.
\end{matrix} ight.


The set indicator function of the Intersection of sets is the minimum function of the indicator functions
:\mathbf{1}_{A\cap B}(x) = \min\{\mathbf{1}_A(x),\mathbf{1}_B(x)\}
The set indicator function of the Union of sets is the maximum function of the indicator functions
  :<math>A \sum_{x\in X} \mathbf{1}_{A}(x)</math>
  :<math>\ \{1,1\} 2 </math>



The constant term ''k''&2·log(2) vanishes by differentiation. The terms ··· vanish in the limit. So for the Standard Normal Distribution , having mean 0 and Standard Deviation 1, the derivative of the cumulant generating function is simply ''g'' '(''t'') = ''t'' . For the Normal Distribution having mean μ and standard deviation σ, the derivative of the cumulant generating function is ''g'' '(''t'') = μ+σ&2·''t'' .

See also Random Variable .


FREE COMMUTATIVE MONOIDS

There is a connection with the Monoid on a set ''X'' can be taken to be the set of finite multisets with elements drawn from ''X'', with the obvious addition operation. Such monoids are also known as (finite) formal sums of elements of ''X'' with natural
coefficents. Compare Free Abelian Group .


REFERENCES