Search results
Results from the WOW.Com Content Network
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:
Infinite sets are so common, that when one considers finite sets, this is generally explicitly stated; for example finite geometry, finite field, etc. Fermat's Last Theorem is a theorem that was stated in terms of elementary arithmetic , which has been proved only more than 350 years later.
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.
Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T 1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable ...
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus
2D-plot: As a generalization of a Boolean matrix, a relation on the –infinite– set R of real numbers can be represented as a two-dimensional geometric figure: using Cartesian coordinates, draw a point at (x,y) whenever (x,y) ∈ R. A transitive [c] relation R on a finite set X may be also represented as
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.
This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form {0, 1, 2, ..., n−1} for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection.