Given a space X and a subset A of X, we say that A is dense in X if the intersection of every open set of X with A is non-empty. E.g., the rational numbers are dense in the real numbers.
dense linear order
A set X is said to be a dense linear order if it is a linear order under a relation “ < ” and such that for all distinct x and z with x < z, there is a y in X such that x < y and y < z. The canonical example of a dense linear order is the set Q of rational numbers.
A set is denumerable if it is infinite and countable.
Given a set X, the derived set of X is the set of accumulation points of X. The second derived set is the derived set of the derived set, and so on.
Cf. Cantor-Bendixson Theorem.
Geometry: A diameter of a circle (or sphere) is a line containing the center and with endpoints on the perimeter (resp. surface).
Analysis: Given a set X in a metric space, the diameter of X is the supremum of the distances between all pairs of points of X.
Graph Theory: The diameter of a given graph G is the maximum, over all pairs of vertices u, v of G that are in the same connected component of G, of the distance between u and v. In other words, it is the greatest distance between two vertices on the graph.
The difference of two numbers m and n, with n > m, is the number which when added to m yields n. For example, the difference of 3 and 5 is 2.
Set Theory: The difference of two sets A and B, denoted either as AB or as A - B, is the set of elements of A that are not in B.
General: So-called “Discrete Mathematics” consists of those branches of mathematics which are concerned with the relations among fixed rather than continuously varying quantities, e.g., combinatorics and probability.
Topology: A topology on a set X is discrete if every subset of X is open, or equivalently if every one-point set of X is open.
Two sets are disjoint if they have empty intersection.
A union of sets which are disjoint.
A set of points consisting of a circle together with its interior points. The set consisting only of the interior points of a circle is called an open disk.
The distance between two points in a space is given by the length of the geodesic joining those two points. In Euclidean space, the geodesic is given by a straight line, and the distance between two points is the length of this line. The distance between two points a and b on a real number line is the absolute value of their difference, i.e., d(a, b) = |a - b|. In two (or more) dimensions, the distance is given by the (generalized) Pythagorean theorem, i.e., in a Cartesian coordinate system of n dimensions, where a = (a1, ... ,an) and b = (b1, ... ,bn), the distance d(a, b) is given by
The concept of distance may be generalized to more abstract spaces – such a distance concept is referred to as a metric.
Graph Theory: The length of the shortest path between two vertices of a graph. If there is no path between two vertices, their distance is defined to be infinite. The distance between two vertices v and u is denoted by d(v, u). In a connected graph, distance is a metric.
See distributive property.
A lattice is called distributive if for all elements x, y, and z of the lattice we have x (y z) = (x y) (x z) and x (y z) = (x y) (x z).
An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac.
Cf. commutative, associative.
To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.
In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a.
A number that is being divided.
A number that is dividing another.
A polyhedron having twelve faces.
The faces of a regular dodecahedron are regular pentagons.
Cf. Platonic solid.
General: A universe of discourse, that is, the class of objects under consideration. Functions and relations: The domain of a function (relation) is the set of elements which the function (relation) maps to its range set.
See Euler number.
General: A line formed by the intersection of two planes. In a 3-dimensional figure (such as a polyhedron), the line or curve where two faces or surfaces meet.
Graph Theory: One of two kinds of entities in a graph. Restricted to being incident on exactly two vertices.