mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle...

mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself...

or even other sets. A set may be finite or infinite, depending whether the number of its elements is finite or not. There is a unique set with no elements...

of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set. The notation for...

cardinality: Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a finite set. Any set X that has the same...

exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists)...

cofinite subset of a set X {\displaystyle X} is a subset A {\displaystyle A} whose complement in X {\displaystyle X} is a finite set. In other words, A...

proper class. A finite family of subsets of a finite set S {\displaystyle S} is also called a hypergraph. The subject of extremal set theory concerns...

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive...

sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the empty sum) is zero. The...

(first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least...

A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line...

cardinality of set B and set C are both 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of...

condition. In other contexts, only chains that are finite sets are considered. In this case, one talks of a finite chain, often shortened as a chain. In this...

term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied...

z}, {y, z}, {x, y, z}}. If S is a finite set with the cardinality |S| = n (i.e., the number of all elements in the set S is n), then the number of all the...

whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as...

fuzzy set A = ( U , m ) {\displaystyle A=(U,m)} . For a finite set U = { x 1 , … , x n } , {\displaystyle U=\{x_{1},\dots ,x_{n}\},} the fuzzy set ( U ...

A filter subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently...

unique. The finitely indexed discrete sets are just the finite sets. In particular, finitely indexed subsets of ω {\displaystyle \omega } are finite. Taking...

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical...

mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable...

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which...

will only be a finite number of mathematicians in the course of our (presumably finite) civilization. Hence there is only a finite set of whole numbers...

an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton (FSA) that maps between two sets of symbols. An...

deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite-state machine (DFSM), or deterministic finite-state...

used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences...

\infty } or − ∞ {\displaystyle -\infty } ). Every finite set function must have a finite mass. A set function μ {\displaystyle \mu } on F {\displaystyle...

of the existence of finite families of open sets that "cover" the space, in the sense that each point of the space lies in some set contained in the family...