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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a partition of a set is a grouping of its elements into
non-empty #REDIRECT Empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

non-empty
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
s, in such a way that every element is included in exactly one subset. Every
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a
setoid In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, typically in
type theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
and
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
.


Definition and Notation

A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a
disjoint union In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

disjoint union
of the subsets). Equivalently, a
family of sets In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are ...
''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
(that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also
collectively exhaustive events In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expres ...
and
cover (topology) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. * The
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of any two distinct sets in ''P'' is empty (that is (\forall A,B \in P)\; A\neq B \implies A \cap B = \emptyset). The elements of ''P'' are said to be
pairwise disjoint Two disjoint sets. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical anal ...
. The sets in ''P'' are called the ''blocks'', ''parts'', or ''cells'', of the partition. If a\in X then we represent the cell containing ''a'' by /math>. That is to say, /math> is notation for the cell in ''P'' which contains ''a''. Every partition, ''P'', may be identified with an equivalence relation on ''X'', namely the relation \sim_P such that for any a,b\in X we have a\sim_P b if and only if a\in /math> (equivalently, if and only if b\in /math>). The notation \sim_P evokes the idea that the equivalence relation may be constructed from the partition. Conversely every equivalence relation may be identified with a partition. This is why it is sometimes said informally that "an equivalence relation is the same as a partition". If ''P'' is the partition identified with a given equivalence relation \sim, then some authors write P = X/\sim. This notation is suggestive of the idea that the partition is the set ''X'' divided in to cells. The notation also evokes the idea that, from the equivalence relation one may construct the partition. The rank of ''P'' is , if ''X'' is
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
.


Examples

*The empty set \emptyset has exactly one partition, namely \emptyset. (Note: this is the partition, not a member of the partition.) *For any non-empty set ''X'', ''P'' = is a partition of ''X'', called the trivial partition. **Particularly, every
singleton set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
has exactly one partition, namely . *For any non-empty
proper subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''A'' of a set ''U'', the set ''A'' together with its
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
form a partition of ''U'', namely, . *The set has these five partitions (one partition per item): ** , sometimes written 1 , 2 , 3. ** , or 1 2 , 3. ** , or 1 3 , 2. ** , or 1 , 2 3. ** , or 123 (in contexts where there will be no confusion with the number). *The following are not partitions of : ** is not a partition (of any set) because one of its elements is the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
. ** is not a partition (of any set) because the element 2 is contained in more than one block. ** is not a partition of because none of its blocks contains 3; however, it is a partition of .


Partitions and equivalence relations

For any
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on a set ''X'', the set of its
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es is a partition of ''X''. Conversely, from any partition ''P'' of ''X'', we can define an equivalence relation on ''X'' by setting precisely when ''x'' and ''y'' are in the same part in ''P''. Thus the notions of equivalence relation and partition are essentially equivalent. The
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
guarantees for any partition of a set ''X'' the existence of a subset of ''X'' containing exactly one element from each part of the partition. This implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class.


Refinement of partitions

A partition ''α'' of a set ''X'' is a refinement of a partition ''ρ'' of ''X''—and we say that ''α'' is ''finer'' than ''ρ'' and that ''ρ'' is ''coarser'' than ''α''—if every element of ''α'' is a subset of some element of ''ρ''. Informally, this means that ''α'' is a further fragmentation of ''ρ''. In that case, it is written that ''α'' ≤ ''ρ''. This ''finer-than'' relation on the set of partitions of ''X'' is a
partial order upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to s ...
(so the notation "≤" is appropriate). Each set of elements has a
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and a
greatest lower bound are equal. Image:Supremum illustration.svg, 250px, A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond). In mathematic ...
, so that it forms a lattice, and more specifically (for partitions of a finite set) it is a
geometric latticeIn the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, ...
.. The ''partition lattice'' of a 4-element set has 15 elements and is depicted in the
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a Graph drawing, drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' on ...
on the left. Based on the
cryptomorphismIn mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. This word is a play on the many morphisms in mathematics, but "cryptomorphism" is only ver ...
between geometric lattices and
matroid In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most si ...
s, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the
atoms An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...
of the lattice, namely, the partitions with n-2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of a
complete graph In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

complete graph
. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; in graph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of the subgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the lattice of flats of the
graphic matroid In the mathematical theory of Matroid theory, matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the tree (graph theory), forests in a given finite undirected graph. The dual matroid ...
of the complete graph. Another example illustrates the refining of partitions from the perspective of equivalence relations. If ''D'' is the set of cards in a standard 52-card deck, the ''same-color-as'' relation on ''D'' – which can be denoted ~C – has two equivalence classes: the sets and . The 2-part partition corresponding to ~C has a refinement that yields the ''same-suit-as'' relation ~S, which has the four equivalence classes , , , and .


Noncrossing partitions

A partition of the set ''N'' = with corresponding equivalence relation ~ is noncrossing if it has the following property: If four elements ''a'', ''b'', ''c'' and ''d'' of ''N'' having ''a'' < ''b'' < ''c'' < ''d'' satisfy ''a'' ~ ''c'' and ''b'' ~ ''d'', then ''a'' ~ ''b'' ~ ''c'' ~ ''d''. The name comes from the following equivalent definition: Imagine the elements 1, 2, ..., ''n'' of ''N'' drawn as the ''n'' vertices of a regular ''n''-gon (in counterclockwise order). A partition can then be visualized by drawing each block as a polygon (whose vertices are the elements of the block). The partition is then noncrossing if and only if these polygons do not intersect. The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in
free probabilityFree probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of statistical independence, independence, and it is connected with free p ...
theory. These form a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.


Counting partitions

The total number of partitions of an ''n''-element set is the
Bell number In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, ...
''Bn''. The first several Bell numbers are ''B''0 = 1, ''B''1 = 1, ''B''2 = 2, ''B''3 = 5, ''B''4 = 15, ''B''5 = 52, and ''B''6 = 203 . Bell numbers satisfy the
recursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the scientific study of language, meaning tha ...

recursion
: B_=\sum_^n B_k and have the
exponential generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...
:\sum_^\infty\fracz^n=e^. The Bell numbers may also be computed using the
Bell triangleImage:BellNumberAnimated.gif, Construction of the Bell triangle In mathematics, the Bell triangle is a triangle of numbers analogous to Pascal's triangle, whose values count Partition of a set, partitions of a set in which a given element is the larg ...
in which the first value in each row is copied from the end of the previous row, and subsequent values are computed by adding two numbers, the number to the left and the number to the above left of the position. The Bell numbers are repeated along both sides of this triangle. The numbers within the triangle count partitions in which a given element is the largest singleton. The number of partitions of an ''n''-element set into exactly ''k'' (non-empty) parts is the
Stirling number of the second kind In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
''S''(''n'', ''k''). The number of noncrossing partitions of an ''n''-element set is the
Catalan number In combinatorial mathematics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is ...

Catalan number
:C_n=.


See also

*
Exact coverIn mathematics, given a collection S of subsets of a set ''X'', an exact cover is a subcollection S^ of S such that each element in X is contained in ''exactly one'' subset in S^. One says that each element in X is covered by exactly one subset in S^ ...
*
Block design In combinatorial mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, ...
*
Cluster analysis Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of ...
*
Weak ordering The 13 possible strict weak orderings on a set of three elements . The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy. In mathematics, especially order theory, a weak ordering ...
(ordered set partition) * Partial equivalence relation *
Partition refinementIn the design of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the ...
* List of partition topics *
Lamination (topology) In topology, a branch of mathematics, a lamination is a : * "topological space partition of a set, partitioned into subsets" * decoration (a structure or property at a point) of a manifold in which some subset of the manifold is partitioned into sh ...
* Rhyme schemes by set partition


Notes


References

* * {{Authority control
Basic concepts in set theory{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...
Combinatorics Set families