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 distance – equivalence relation 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 byThe 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   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). distributive property   An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac. Cf. commutative, associative. divide   To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a. dividend   A number that is being divided. divisor   A number that is dividing another. dodecahedron   A polyhedron having twelve faces.The faces of a regular dodecahedron are regular pentagons.Cf. Platonic solid. 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. dot product   See scalar product. e   See Euler number. edge   General: A line formed by the intersection of two planes. In a 3-dimensional figure (such as a polyhedron), the line or curve where two faces or surfaces meet.Graph Theory: One of two kinds of entities in a graph. Restricted to being incident on exactly two vertices. edge set   The set of edges of some graph. For a graph G, the edge set of G is denoted by E(G), or, if there is no ambiguity as to the graph in question, simply by E. ellipse   The locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.Like the hyperbola and parabola, the ellipse is a conic section. Related article: Conics 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. epimorphism   A morphism f from X to Y is called an epimorphism when it is surjective, that is, when to each element y of Y there corresponds an x in X such that f(x) = y.Cf. monomorphism. equicontinuous   Let C[a,b] denote the space of all continuous functions on the real interval [a,b]. A subset S of C[0,1] is called equicontinuous at x in [a,b] if for any e greater than zero there is some d greater than zero such that for every function f in S we have: equilateral triangle   A triangle with equal sides and equal angles. equipotent   Two sets are equipotent if there exists a function between them that is bijective, that is, which is “one-to-one” and “onto.”Cf. cardinal. equivalence relation   An equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive. An equivalence relation on X gives rise to (and is determined by) a partition on X.Cf. congruence relation. distance – equivalence relation
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