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countable – derived set
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.
cross product
See vector product.
csc
See cosecant.
cube
A regular polyhedron having six square faces.
More generally, an n-dimensional cube in the first quadrant of a Euclidean space with one vertex at the origin is given by the collection of all n-tuples of the form (e1,e2, ... , en), with each ei an element of the set {0,1}.
Cf. platonic solid.
Related article: Platonic Solids
cub set
Closed, unbounded set. See closed set.
cycle
Algebra: See permutation.
Graph Theory: A graph C whose vertices and edges can be numbered {v1, v2, ..., vn} and {e1, e2, ..., en}, respectively, for some n 1, such that ei is incident on vi and vi + 1 for every i < n, and en is incident on vn and v1. A cycle with n vertices is denoted by Cn. For a given graph G, a cycle of G (or a cycle in G) is a subgraph of G that is also a cycle. Cycles are sometimes also called circuits.
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.
Dandelin’s Spheres
ARTICLE
A proof by the 17th century French mathematician Germinal Dandelin of the equivalence of the plane-geometry and conic-section definitions of the ellipse, parabola, and hyperbola. See the article for a full exposition.
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.
deficient number
See: abundant number.
definite integral
See: integral.
degenerate conic
A conic section in which the intersecting plane passes through the vertex.
Related article: Conics
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.
derivative
For a function f of a single real variable x, the derivative is defined as the limit of the difference quotient
provided this limit exists. In practice, the derivative is interpreted as the instantaneous rate of change of the function at x. Graphically, the derivative returns the slope of the tangent at x.
Cf. differentiation rules.
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.
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