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 extended real numbers – homeomorphism extended real numbers   The set of real numbers together with a “point(s) at infinity.” If both a positive and a negative infinity are added, then the resulting space is order equivalent and topologically equivalent (i.e., homeomorphic) to the unit interval [0, 1]. If only a single point at infinity is added, the resulting space is topologically equivalent to the unit circle. fallacy    ARTICLE   An unjustified step in logic, or incorrect form of reasoning, leading to an invalid conclusion. See the article for a complete description. filter   If X is a set (or class) and F is a family of subsets of X, then F is called a filter providedF is closed under intersections, i.e., for any sets A, B in F their intersection is also in F, andF is closed under supersets (upwardly closed), i.e., if A is any set in F and B is any set in X containing A, then B is in F. If X and F are as above, and if in addition for every subset Y of X either Y or its complement is in F, then F is called an ultrafilter (or maximal filter). Equivalently, an ultrafilter is a filter that is not the proper subset of any filter. A free ultrafilter is an ultrafilter which contains the complement of every finite set. If A is any subset of X, then the collection of supersets of A together with all finite intersections of supersets of A is called the filter generated by A. More generally, if U is any family of subsets of X that is closed unter finite intersections, then the family F of subsets which includes precisely U and all of the intersections of supersets of members of U is the filter generated by U, and U is called its filter base. If a filter F is generated by a singleton set, then it is called a principal filter. finite intersection property   A collection of sets has the finite intersection property if every finite subcollection has non-empty intersection. finite measure   See measure. finite set   A set X is finite if there is a natural number n (possibly 0) such that X can be said to have exactly n elements. More formally, a set is finite if it is not bijective with any proper subset of itself. flat pair   A pair of elements {a,b} with no order defined on the elements. Sometimes also called an unordered pair.Cf. ordered pair. function   Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y.More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively. functional calculus   See predicate calculus. Fundamental Theorem of Calculus   Intitively, that the integral and derivative are inverse operatorations on functions. The fundamental theorem is commonly expressed in either one of two formal guises:Cf. Riemann integral. GCH   See: generalized continuum hypothesis. generalized continuum hypothesis   The claim that there is no set of intermediate cardinality between any given set and its power set. The GCH is independent of (i.e., cannot be proved true or false by) the known axioms of set theory.Cf. continuum hypothesis. Related article: Gödel's Theorems Related MiniText: Infinity -- You Can't Get There From Here... geometric series   An infinite series such that the ratio of consecutive terms is constant. Such a series may be denotedwhere r is the common ratio. A geometric series converges if and only if the common ratio is less than 1, in which case the sum of the series is given byCf. arithmetic series, harmonic series. Related article: Series g.l.b.   Abbreviation for greatest lower bound. graph   General: A picture or diagram representing one or more relationships among certain objects or quantities.Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph. Cf. Konigsberg Bridge Problem. greatest lower bound   A lower bound which is greater than or equal to every other lower bound.Cf. Least upper bound. harmonic series   The infinite series whose terms are the reciprocals of the natural numbersThis series diverges, i.e., the sum does not exist. Related article: Series Hausdorff space   A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed. Heine-Borel property   Every closed and bounded subset of the real numbers is compact. Hempel’s Ravens Paradox    ARTICLE   A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition. homeomorphism   A homeomorphism is a bijective, continuous transformation of one topological space onto another whose inverse is also continuous. extended real numbers – homeomorphism
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