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Euclidean space – geodesic
Euclidean space
A space satisfying the axioms of Euclidean geometry. Specifically, a space in which the parallel postulate holds; equivalently, a space in which the distance between points is given by a straight line.
Euler Polyhedron Formula
V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces of a polyhedron or a planar graph.
even function
A real-valued function y = f(x) is even if f(x) = f(–x) for all x in the domain of f. The graph of an even function in the Cartesian plane is symmetric with respect to the y-axis.
Cf. odd function.
existential quantifier
See predicate calculus.
exp
See exponential function.
exponential function
The function ex. We say that an expression is expontiated when it is used as an exponent on e.
More generally, any function which operates on its argument by placing it as an exponent may be called an exponential function. Example: f(x) = 2x.
Cf. Euler number.
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.
figure
General: A drawing, picture, or illustration, usually accompanying a text description. Also synonymous with digit or numeral.
Geometry: A graphical (visual) representation of points, lines, curves, surfaces, solids, or regions. The word “figure” may also be used to refer to the abstract object or set of points thus represented.
Cf. plane figure.
filter
If X is a set (or class) and F is a family of subsets of X, then F is called a filter provided- F is closed under intersections, i.e., for any sets A, B in F their intersection is also in F, and
- F 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 graph
A graph whose vertex and edge sets are finite. In all papers, theorems, conjectures, textbooks, discussions, and interactions of graph theory all graphs considered are always finite, unless explicitly specified otherwise.
Cf. infinite graph.
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...
geodesic
A curve on a space is a geodesic if, given any two points on the curve, the segment of the curve joining the points is the shortest such curve. In Euclidean space, geodesics are straight lines. On a sphere, the geodesics are the great circles of the sphere.
Graph Theory: A shortest path between two vertices is sometimes called a geodesic.
Cf. distance.
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