Partition of a setIn mathematics, a partition of a set is a grouping of its elements into non-emptysubsets, in such a way that every element is included in exactly one subset. Every equivalence relation 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, typically in type theory and proof theory. Definition and notationA partition of a set X is a set of non-empty subsets of X such that every element ...
Cardinal numberIn mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set A {\displaystyle A} is generally denoted by | A | {\displaystyle \vert A\vert } , with a vertical bar on each side, though it may also be denoted by A {\displaystyle A} , card ( A ) , {\displaystyle \operatorname {card} (A),} or # A . {\displaystyle \#A.} Cardinality is defined in terms of bij...
SubsetIn mathematics, a setA is a subset of a set Bif and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. When quantified, A⊆B{\displaystyle A\subse...
Axiom of choiceIn mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one element chosen from each set, even if the collection is infinite. Formally, the axiom establishes existence rather than a construction; it states that for every set I {\displaystyle I} and every I {\displaystyle I} -indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, th...
Axiom of determinacyIn mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a...
Infinite setIn set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. PropertiesThe set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natura...
Ramsey's theoremIn combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. As the simplest example, consider two colours (say, blue and red). Let r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique ...
Continuum (set theory)In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by c {\displaystyle {\mathfrak {c}}} . Georg Cantor proved that the cardinality c {\displaystyle {\mathfrak {c}}} is larger than the smallest infinity, namely, ℵ 0 {\displaystyle \aleph _{0}} . He also proved that c {\displaystyle {\mathfrak {c}}} is equal to 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}\!} , the cardinality of the power set of the natural numbers. The cardinality of...
Martin's axiomIn the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, c {\displaystyle {\mathfrak {c}}} , behave roughly like ℵ 0 {\displaystyle \aleph _{0}} . The intuition behind this can be under...
CombinatoricsCombinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability t...
Tree (set theory)In set theory, a tree is a partially ordered set ( T , < ) {\displaystyle (T,<)} such that for each t ∈ T {\displaystyle t\in T} , the set { s ∈ T : s < t } {\displaystyle \{s\in T:s<t\}} is well-ordered by the relation < {\displaystyle <} . Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.
GraphonIn graph theory and statistics, a graphon (also known as a graph limit) is a symmetricmeasurable function W:[0,1]2→[0,1]{\displaystyle W:[0,1]^{2}\to [0,1]}, that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable random graph models. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons give rise to dense graphs almo...*As an Amazon Associate I earn from qualifying purchases.