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 contrapositive – difference contrapositive   Given a conditional statement, i.e., a statement of the form “if A then B,” or “A implies B,” its contrapositive is “not B implies not A.” A conditional and its contrapositive are logically equivalent. Consequently, in mathematics, to prove a conditional it is a good strategy (and often easier) to prove its contrapositive instead.Cf. inverse statment, converse statement. converse relation   If R is a relation, the relation R´ is called the converse relation of R if whenever xRy we have yR´x. converse statement   Given a conditional, i.e., a statement of the form “if A then B,” or “A implies B,” its converse is “B implies A.” A conditional neither implies nor is implied by its converse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other. convex   Naively, a region of space is convex if the line segement joining any two points of the region lies wholly within it. Thus, a polygon is convex if every line segment joining any two points on its sides lies entirely within the polygon. (This is equivalent to the condition that all its interior angles be less than 180°.)More generally, a region in a real vector space is convex if whenever two points x and y are in the region then so is any point tx + (1 - t)y, where t lies in the interval [0, 1]. See the immediately following entries for additional uses of the descriptor “convex.”Cf. concave. convex set   A subset X of a partially ordered set is convex if for any elements a, b in X such that a b, then all elements x satisfying a x b are also in X. 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. 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. 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 cutArithmetic 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. 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. diameter   Geometry: A diameter of a circle (or sphere) is a line containing the center and with endpoints on the perimeter (resp. surface).Analysis: Given a set X in a metric space, the diameter of X is the supremum of the distances between all pairs of points of X.Graph Theory: The diameter of a given graph G is the maximum, over all pairs of vertices u, v of G that are in the same connected component of G, of the distance between u and v. In other words, it is the greatest distance between two vertices on the graph. difference   The difference of two numbers m and n, with n > m, is the number which when added to m yields n. For example, the difference of 3 and 5 is 2.Set Theory: The difference of two sets A and B, denoted either as AB or as A - B, is the set of elements of A that are not in B. contrapositive – difference
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