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countable – Epimenides Paradox
countable
A set is countable if it is finite, or if it is infinite and bijective to the set of natural numbers (finite ordinals), i.e., if there exists a complete one-to-one mapping of the set in question onto the set N. Sets that are both countable and infinite are sometimes called denumerable. Georg Cantor proved that sets may be uncountably infinite, for example the set of real numbers.
Related MiniText: Infinity -- You Can't Get There From Here...
countable chain condition
See chain condition.
countably infinite
See countable
cover
Topology: a collection of sets which contains a given set. If the sets in the covering collection are open sets, the cover is called an open cover.
Partially ordered sets: If x and y are elements of a partially ordered set such that x  y, and such that there is no z such that x z y, then we say that x covers y.
Graph Theory: An edge or vertex of a graph is said to cover (verb) those vertices or edges, respectively, that it is incident on. A set of edges or vertices is said to cover any vertex or edge covered by any element of that set. A set of edges or vertices that covers all the vertices or edges, respectively, of the graph is called a cover (noun), usually with a specification of whether it consists of edges or vertices. See edge cover, vertex cover.
cub set
Closed, unbounded set. See closed set.
D-system
Set Theory: A family A of sets is called a D-system (also called a quasi-disjoint family) if and only if there is a fixed set r, called the root of the D-system, such that for any two elements x and y of A, x  y = r. It is a theorem that every uncountable family of finite sets has a sub-family which forms a D-system.
Cf. almost disjoint.
Dedekind cut
A partition of the set of rational numbers into two subsets A and B, such that every element of A is less than every element of B. The set of real numbers can be formally defined as the set of all Dedekind cuts; e.g., we can take the real number square root of 2 to be the cut
Arithmetic on real numbers is then defined in terms of operations on cuts. For example, if (A1, B1) and (A2, B2) are Dedekind cuts defining real numbers x and y, then we define the inequality x < y to mean that there is an element of A2 that is not an element of A1, and we define x + y to be the real number corresponding to the Dedekind cut (A, B) where the elements of A are all those rational numbers obtained by adding an element of A1 to an element of A2.
The set of irrational numbers can then be defined as the set of all real numbers that cannot be represented as a ratio of integers.
degree
Geometry: A unit of measure of angles, denoted by ° . A complete circle contains 360°, a straight line 180°, and a right angle 90°. Compare radians.
Algebra: The degree of a polynomial is the highest sum of exponents appearing in any term with a non-zero coefficient.
Graph Theory: The degree of a vertex is the number of edges incident on that vertex, counting loops twice. The degree of an edge is an unordered pair of the degrees of its two ends.
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.
derived set
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.
diameter
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.
disjoint
Two sets are disjoint if they have empty intersection.
disjoint union
A union of sets which are disjoint.
disk
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.
Cf. neighborhood.
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 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).
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.
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.
Epimenides Paradox
See Liar Paradox.
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