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In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. [4] If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers , and the class of all sets, are proper classes in many formal systems.
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
S is a finite set. Not infinite; Finite; Empty semigroup: S = No Trivial semigroup: Cardinality of S is 1. Infinite; Finite; Monoid: 1 ∈ S; No Gril p. 3 Band (Idempotent semigroup) a 2 = a; Infinite; Finite; C&P p. 4 Rectangular band: A band such that abca = acba; Infinite; Finite; Fennemore: Semilattice: A commutative band, that is: a 2 = a ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
3. A transitive set or class is a set or class such that the membership relation is transitive on it. 4. A transitive model is a model of set theory that is transitive and has the usual membership relation tree 1. A tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation < 2.
Just as for finite sets, the theory makes further definitions which allow us to consistently compare two infinite sets as regards whether one set is "larger than", "smaller than", or "the same size as" the other. But not every intuition regarding the size of finite sets applies to the size of infinite sets, leading to various apparently ...
Any family of sets without repetitions is a subclass of the proper class of all sets (the universe). Hall's marriage theorem, due to Philip Hall, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a system of distinct representatives.