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dense – empty set
dense
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.
Related MiniText: Number -- What Is How Many?
denumerable
A set is denumerable if it is infinite and countable.
difference
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.
discrete
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.
disjoint
Two sets are disjoint if they have empty intersection.
disjoint union
A union of sets which are disjoint.
distance
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.
distributive
See distributive property.
distributive lattice
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).
distributive property
An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac.
Cf. commutative, associative.
divide
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.
dividend
A number that is being divided.
divisor
A number that is dividing another.
dodecahedron
A polyhedron having twelve faces.
The faces of a regular dodecahedron are regular pentagons.
Cf. Platonic solid.
domain
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.
e
See Euler number.
edge
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.
edge set
The set of edges of some graph. For a graph G, the edge set of G is denoted by E(G), or, if there is no ambiguity as to the graph in question, simply by E.
ellipse
The locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.
Like the hyperbola and parabola, the ellipse is a conic section.
Related article: Conics
empty set
The unique set having no elements, generally denoted by a circle with a forward slash through it, or by an empty pair of braces.
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