Information Technology Reference
In-Depth Information
a set, while a multiset can be seen as a function from a set to a subset of natural
numbers (the multiplicities assigned to the elements of the set).
A sequence
α
of length
n
over
A
is a function:
α
:
{
1
,
2
,...
n
}→
A
A multiset
X
over
A
, where any element occurs at most
n
times, is a function:
X
:
A
→{
0
,
1
,
2
,...
n
}.
A very useful notation which assumes many meanings, in dependence on the
context where it occurs, is the absolute value sign
||
. When it is applied to a (finite)
set,
X
,then
means the number of elements of
X
, also called
cardinality
(in set
theory it extends also to infinite sets, providing
transfinite cardinal numbers
). For a
sequence
|
X
|
α
, the expression
|
α
|
denotes the
length
of
α
, while for a (finite) multiset
|
|
X
, notation
X
denotes its
size
, that is, the sum of the multiplicities of all elements
occurring in
X
.
Given an order among the elements of a finite set
A
of cardinality
n
, any finite
multiset
X
over
A
, is completely identified by the numeric sequence, of length
n
, of the multiplicities that the elements of
A
have in
X
(according to the order
fixed over
A
). In this sense, finite multisets coincide with finite sequences of
numbers.
The main principles of aggregation and selection mechanisms, on which life is
based, rely on the basic discrete mathematical structures of sequences and multisets.
We claim that many choices of life for realizing its strategies of expansion and
development are intrinsic to the informational logic of these fundamental structures.
1.4
Chemistry Multisets
Any molecule is a multiset of atoms providing a stable physical structure. In
molecules, multiplicities are indicated by indexes, thus
CO
2
has the same meaning
of
C
+
2
O
. Chemical substances are multisets of molecules. In fact, 100 molecules
of
CO
2
correspond to a multiset of molecules (100 copies of the same molecule).
A quantity of 12 grams of Carbon dioxide
CO
2
is a multiset of about 6
10
23
.
2
×
of
CO
2
molecules.
Chemistry suggests some natural operation on multisets. Firstly, given two mul-
tisets
X
,
Y
over a set
A
,their
multiset sum
X
+
Y
can be defined by setting that, for
any
a
∈
A
, the multiplicity of
a
in
X
+
Y
is given by:
(
X
+
Y
)(
a
)=
X
(
a
)+
Y
(
a
)
