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 closed – conditional statement 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 operator   If X is a set, then a function C from P(X) into P(X) (i.e., a function on the power set of X) is called a closure operator providedY is contained in C(Y) for every subset Y of X, C(C(Y)) = C(Y) for every subset Y of X, andIf Y and Z are both subsets of X, with Y a subset of Z, then C(Y) is a subset of C(Z). Closure operators induce closed set systems. closure property   A property of subsets of a set X is a closure property if X has the property and the intersection of any subsets of X having the property also has the property. club set   Closed, unbounded set. See closed set. coefficient   See polynomial. cofinality   A function f which maps an ordinal a into an ordinal b is said to map a cofinally if the range of f is not bounded in b. (I.e., for every b in b there is an a in a such that f(a) b.) The cofinality of an ordinal b is the least ordinal a such that there is a cofinal map of a into b. combination   A subselection of a set of r elements from a set of n elements. The number of such combinations, i.e., the number of ways in which r elements may be chosen from a set of n elements, is given by the formulaThis operation is also sometimes denoted by nC r, and is read “n choose r.”Cf. permutation, factorial. common logarithm   A logarithm with base 10. commutative   An operation “ · ” on elements of a set A is commutative if for all elements a, b in A, a · b = b · a.Cf. associative, distributive property. commutative property   A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative. complement   The complement of a set A is the set of all elements that are not elements of A.Graph Theory: The complement of a simple graph G with vertex set V is the simple graph Gc, which also has vertex set V, and in which two vertices are adjacent if and only if they are not adjacent in G. complementary angles   Two angles are complementary if they add up to a right angle. complete   Analysis: A metric space X is complete if every Cauchy sequence in X converges in X.Logic: a system of axioms for a mathematical theory is complete if every theorem in the theory is deducible from the axioms. Gödel's incompleteness theorem states that any axiom system which includes or allows the operations of arithmetic is necessarily incomplete.Set Theory: If F is a filter on a set X and k is a regular, uncountable cardinal, then we say that F is k-complete (k-closed) if AF for every AF with |A| < k. Every filter is w-complete. If k is the first uncountable cardinal (1), then F is called countably complete. Related article: Gödel's Theorems complete lattice   A lattice X is complete if every subset of X has a least upper bound and a greatest lower bound in X. complete ordered field   A linearly ordered field in which every subset with an upper bound has a least upper bound. Every complete ordered field is isomorphic to the set of real numbers. complex number   An element of the set C of numbers of the form a + bi, where a and b are real numbers and i 2 = -1. The number i is called the imaginary number. concave   A region of space is concave if there are two points of the region such that a line joining the two points is not entirely contained within the region. In particular, a polygon is concave if any of its interior angles is greater than 180°.Cf. convex. conditional statement   A statement of the form “if A then B,” or “A implies B.” A conditional is equivalent to its contrapositive, “not B implies not A.” See also: inverse statement and converse statement. closed – conditional statement
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