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 coefficient – constant 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. commutative ring   A ring in which the multiplication operation is commutative. compact   Topology: In a topological space, a set E is compact if every open covering of E has a finite subcover, i.e., a finite subcollection which also covers E. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. E is called s-compact if there exists a sequence of compact sets {Ci} such that E is contained in their union.Cf. locally compact, Bolzano-Weierstrass property, Heine-Borel property.Set Theory: A cardinal k is called weakly compact if it is uncountable and Equivalently, k is weakly compact if it is strongly inaccessible and there are no k-Aronszajn trees.Lattices: an element a of a lattice L is called compact if whenever a is dominated by the join of a subset X of L then a is dominated by the join of a finite subset of X. Symbolically: 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. Gφdel'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 measure   See measure. 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. concave function   A function is concave if the chord connecting any two points of its graph lies entirely below the graph.Cf. convex function. 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. congruence relation   Given a structure X (i.e., X is a space, algebra, etc.), then an equivalence relation  ~  on X is called a congruence relation if it is preserved by every defined operation on X. That is, for any x, y in X, if x ~ y and f is defined on X, then f(x) ~ f(y). connected   A topological space X is connected if there are no two open sets of X whose union is X and whose intersection is empty. constant   An unvarying quantity, usually represented notationally by an alphabetic letter such as k, c, etc. coefficient – constant
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