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equivalence relation – Fundamental Theorem of Algebra
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.
Euclidean geometry
The theory of geometry arising from the five postulates (axioms) of Euclid: - That any two points determine a line;
- That any line may be extended indefinitely;
- That a circle may be drawn with any center and radius;
- That all right angles are equal to one another;
- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Euclidean geometry is characterized by the 5th postulate, also called the “parallel’s postulate,” which may be restated as; given a line and a point not on the line, exactly one line may be drawn through the given point parallel to the given line. Theories of geometry which reject the 5th postulate form the subject of non-Euclidean geometry. Euclid recorded his theory of geometry in his 13-book opus titled Elements, c. 300 b.c., and his theory was considered the only “true” theory of geometry until the 18th century, when non-Euclidean geometry was discovered independently by Nikolai Lobachevsky, Janos Bolyai, and Karl Friedrich Gauss. Euclid’s axioms were found to be deficient in the 19th century, and Euclidean geometry was re-axiomatized by David Hilbert with the addition of a between-ness axiom, and the relegation of the concepts of point and line, which Euclid had defined, to the status of undefined terms.
Euler number
The transcendental number e, approximately 2.71828.... It may be defined as the limit
or as the infinite sum
The function ex is called the exponential function, and has the property that it is its own derivative.
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.
exponent
Also called index, a number or expression applying to another number or expression, called the base, and indicating that the base is to be “raised to the power of” the exponent. For integers, this corresponds to the operation of repeated self-multiplication the number of times specified by the exponent.
The exponent is written to the right of, and superscripted to, the base.
Cf. rational exponent, laws of exponents.
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.
factor
The factors of a natural number n are those whole numbers which divide it evenly. Since every number is divisible by itself and 1, neither 1 nor n is considered a proper factor of n.
More generally, if an expression can be written as a product of other expressions, then those other expressions are its factors. For instance, a polynomial with rational coefficients may always be factored into a product of first and/or second degree polynomial factors.
A number (or other expression) with no proper factors is called prime.
Graph Theory: A spanning subgraph of a given graph having at least one edge. In many contexts, it is interesting to determine whether some graph G can be decomposed into the edge-disjoint union of factors with some prescribed property. Such a decomposition is called a factorization of G. Often, the property in question is regularity of degree k. In this case, the factors are called k-factors, and the factorization a k-factorization. If G has a k-factorization, it is called k-factorable.
factorial
An operation on natural numbers, denoted n! and given by n! = 1 × 2 × 3 × ... × n. Thus 3! = 1 × 2 × 3 = 6, and 4! = 24. By convention, 0! = 1 and 1! = 1.
factorization
The process of factoring, that is, of finding proper factors.
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 intersection property
A collection of sets has the finite intersection property if every finite subcollection has non-empty intersection.
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 Algebra
The field of complex numbers is
algebraically closed, that is, any polynomial with complex coefficients has a complex root.
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