BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry uncountable – whole number 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 byIf 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 byWhen 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   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. vector product   The vector product (also called cross product) of two vectors u and v, denoted u × v and called “u cross v,” is a vector w whose magnitude (length) is the product of the magnitudes of u and v and the sine of the angle between them, and which points in a direction perpendicular to the plane containing u and v so as to form a right-handed system, as in the figure.Note that the directedness of the vector product implies that it is not commutative.Cf. scalar product. vector space   A structure consisting of two kinds of elements called scalars and vectors, with operations of addition of pairs of scalars or pairs of vectors, and multiplication of pairs of scalars or a scalar and a vector. The vectors form an Abelian group under addition, and the scalars form a field under their operations, and the vector space is said to be over that field.If the scalar field is the real numbers or the complex numbers and the vectors are in n-dimensional real or complex space, then the space is called an n-dimensional real or complex vector space accordingly. Multiplication of vectors by scalars is associative with scalar multiplication, and distributive over both scalar addition and vector addition. Symbolically, for scalars a, b, and vectors u, v,Vector spaces are usually denoted by V, and it is conventional to write the scalar on the left of a scalar multiplication. When there is any possibility of confusion, the vectors of a vector space are usually specially marked, either by drawing a (right pointing) arrow over them or by writing them in bold face. Cf. module. 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. uncountable – whole number
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