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ultrafilter – Zeno of Elea
ultrafilter
A filter F on a set X is called an ultrafilter if for every subset Y of X, either Y is in F or the complement of Y is in F.
uncountable
A set is uncountable (uncountably infinite) if it is infinite but not countable, i.e., no complete one-to-one match-up of the set with the set of natural numbers (finite ordinals) can be performed. Georg Cantor proved that the set of real numbers is uncountably infinite. (This is sometimes called the “non-denumerability of the continuum”).
Related MiniText: Infinity -- You Can't Get There From Here...
uncountably infinite
See uncountable.
union
The union of two sets A and B is the set consisting of those elements which are in A or in B or in both, and is denoted by
If the union is taken over a family of sets {Ai}i = 1, 2, ..., n, then it is the set consisting of those elements that are in at least one of the sets in the collection.
Sometimes the word “union” is used to indicate the sumset of a set A, and is then loosely described as “union A.” In this latter case the union is understood to be the union over the elements of A, and is denoted by
When considering an algebra of sets, the union of two or more sets is sometimes called the join of the sets.
Cf. intersection.
unit circle
A circle with radius 1.
unit interval
The interval on the real number line from 0 to 1, inclusive.
unit square
The set of points of the Cartesian plane with domain and range values in the unit interval, that is the square region with vertices (0, 0), (0, 1), (1, 0), and (1, 1), including its boundary.
universal quantifier
See predicate calculus.
universal Turing machine
A Turing machine capable of performing, given the appropriate program, the actions of any other Turing machine.
unordered pair
See flat pair.
upper bound
An upper bound of a set with an order relation (such as “ < ”) defined on it is an element which is greater than or equal to every element in the set.
Cf. least upper bound.
vector
A quantity having two components; a magnitude component and a direction component. In n-dimensional Euclidean space, a vector is representable by an ordered tuple (a1, a2, a3, ... an) whose elements are called the components of the vector. In this case the magnitude of the vector is given by the square root of the sum of the squares of the components of the vector.
vertex
Geometry: In a plane figure, a point which is a common end-point for two or more lines or curves.
Graph Theory: One of two kinds of entities in a graph.
Cf. edge.
vertex set
The set of vertices of some graph. For a graph G, the vertex set of G is denoted by V(G), or, if there is no ambiguity as to the graph in question, simply by V.
Von Neumann Heirarchy
ARTICLE
A construction in set theory giving rise to the class of ordinals. One begins with the empty set, and successor sets are obtained by forming the union of the current set and the set containing the current set; at limit stages, take unions. See the article for a detailed exposition.
weakly connected digraph
A directed graph every two vertices of which are connected by a semipath. Also called weak for short. A directed graph is weak if and only if it has a spanning semiwalk.
Cf. unilaterally connected digraph, strongly connected digraph, connected graph.
well-founded
In set theory, a collection is well-founded if every subcollection has a least member under the membership relation. For example, the set of natural numbers is well-founded. In ZFC, the foundation axiom asserts this property of all sets. A set which is not well-founded is sometimes called a hyperset.
well-ordered
A set S with a linear order is called well-ordered if every non-empty subset T of S has a least element under the ordering relation.
Cf. well-ordering principle.
well-ordering principle
The assertion that every set can be well-ordered. Equivalent to the Axiom of Choice.
whole number
An element of the set consisting of the number 0 together with the counting numbers, 1, 2, 3, etc.; i.e., the set N of natural numbers with 0 included. Sometimes the term “whole” is meant to refer to negative integers also; the intended meaning should be clear from the context.
Zeno of Elea
Greek philosopher famous for certain paradoxes, including the Paradox of the Arrow, the Paradox of the Tortoise and Achilles, and the Paradox of the Moving Rows. These constitute the earliest known arguments from infinity.
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