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.
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.
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.
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.
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
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.
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.
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.
The assertion that every set can be well-ordered. Equivalent to the Axiom of Choice.
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’s Paradox of the Arrow
One of the famous paradoxes of Zeno of Elea. Consider an arrow in flight. At each moment of time the arrow may be considered to occupy a specific region in space. This poses the following problem: if the arrow is in a particular point of space in any given moment, then how is it moving? And if it is not moving, how does it get from point to point in the passage of time?
Zermelo Fraenkel set theory
The standard set theory in which most mathematics is formalized. Its axioms include the pairing axiom, the power set axiom, the axiom of infinity, the axiom of extensionality, the axiom of replacement, the separation axiom, the union axiom, and the foundation axiom. Abbreviated ZF. When the axiom of choice is assumed, this theory is abbreviated ZFC.
The ring with only one element, its additive identity.
See Zermelo Fraenkel set theory.
Zermelo Fraenkel set theory with the axiom of choice.