BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry cardinality – closure cardinality   Two sets X and Y have the same cardinality if there exists a bijection between them. Specifically, the cardinality of X may be understood as a canonical object meant to represent the proper class of sets to which X is bijective. If the axiom of choice is assumed, then the cardinality of X is the unique cardinal number having the same cardinality as X Carmichael number   A composite integer n which satisfies the congruence an = a modulo n for all integers a. Thus they pass the Fermat test without being prime. Carmichael numbers are characterised by the Korselt condition that they are composite, squarefree and p-1 | n-1 for every prime factor p | n. The first Carmichael number is 561 = 3 × 11 × 17. It was proved in 1994 by Alford, Granville and Pomerance that there are infinitely many Carmichael numbers. Carroll, Lewis   Carroll’s Paradox    ARTICLE   Paradox published by Lewis Carroll as “What the Tortoise Said to Achilles.” See the article for a full exposition.Cf. Charles Lutwidge Dodgson. Cartesian plane   The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane. Cartesian product   For any collection {Ai}, i = 1, 2, 3, ..., n, of sets, the Cartesian productis the set of ordered n-tuples (a1,a2, ... ,an) with a1 an element of A1, a2 an element of A2, etc. The Cartesian product R2 of the set of real numbers is called the Cartesian plane, and in general n-dimensional real space is the Cartesian product Rn. The assertion that the Cartesian product of an infinite collection of non-empty sets is non-empty is equivalent to the axiom of choice. Cauchy sequence   A sequence (x1, x2, x3, ... ) of elements of a metric space X with metric d(x, y) is Cauchy if for any e greater than zero there is some natural number N such thatIn other words, in a Cauchy sequence, the elements eventually become “arbitrarily close together.” If the metric space X is closed, this condition is equivalent to the sequence being convergent. CH   See: continuum hypothesis. chain   If X is a partially ordered set, then a subset Y of X is called a chain if it is totally ordered, that is, if for any two elements a, b of Y, either a b or b a.Cf. antichain. chain condition   For a an infinite cardinal, a partial order P is said to have the a-chain condition if every antichain in P has cardinality not greater than a. If a = w, this is called the countable chain condition, or “c.c.c.” characteristic function   Given a subset E of a space X, the characteristic function cE is defined by cE(x) = 1 if x is in E, and cE(x) = 0 otherwise. All properties of sets and set operations may be expressed by means of characteristic functions. choice, axiom of   See: axiom of choice. chord   A straight line segment connecting two points on a curve or surface. circle   In a plane, the locus of all points equidistant from a given point, called the center. The general equation for a circle in the Cartesian plane is given by (x - h) 2 + (y - k) 2 = r 2, where r is the radius of the circle (distance from the center to the locus of points), and (h, k) are the coordinates of the center.The interior of a circle is referred to as an open disk.A circle is also a conic section; a special case of an ellipse in which the foci coincide. Related article: Conics circumference   Geometry: The distance around a circle in the plane, or around a great circle of a sphere.Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles. class   See proper class. closed   General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.See the following listings for other uses of the word “closed” in mathematics.Topology: A set is topologically closed if it is not open. closed interval   An interval of the real number line (or any other totally ordered set) which includes its endpoints. An interval containing only one of its endpoints is called half-open. Cf. open interval. closed set   Topology: A subset E of a topological space X is closed if X - E (set difference) is open. In a metric space, E is closed if every convergent sequence in E converges in E; equivalently, if every accumulation point of E is in E.Set Theory: If a is a limit ordinal, then a set C contained in a is called closed if and only if for every limit ordinal b less than a, if C b is unbounded in b, then b C. C is called c.u.b. (“cub set” or “club set”) if and only if C is closed and unbounded in a.Cf. stationary set. closed set system   If X is a set (or proper class) and F is a family of subsets of X, then F is called a closed set system providedX is a member of F, andF is closed under arbitrary intersections.Cf. filter. closure   Topology: The closure of a subset E of a topological space is the smallest closed set containing E. It may also be expressed as the union of E with its accumulation points. If E is closed, then it is equal to its closure.Algebra: An algebraic closure of a field F is a field G containing F such that every polynomial with coefficients from F has a root in G. cardinality – closure
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