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algebra – closed
algebra
The term algebra has broadened enormously in modern mathematics from its original meaning as the abstract study of the laws of arithmetic, in which letters or other symbols are used in place of specific numbers in equations or other arithmetic statements. The term algebra should now be understood to denote any set of objects together with a collection of finitary operations defined on it. Thus, we may have an algebra of sets, an algebra of numbers, an algebra of functions, and so on. See the following entries for many particular uses of the words "algebra" and "algebaic."
algebraic function
A function which operates on its variable(s) by addition, subtraction, multiplication, division, raising to a power, or extraction of roots. Compare: transcendental function.
algebraic number
A real or complex number which is the root of a polynomial with rational coefficients. Numbers that cannot be so expressed are called transcendental numbers. The algebraic numbers are countable.
angle
A figure formed by two line segments that extend from a common point. Also refers to the measure of the angle, in degrees or radians, indicating the amount by which one of the line segments must be rotated about the common point to make it coincide with the other segment. The angle between two planes is the angle between two intersecting lines, one lying in each plane and perpendicular to the line of intersection of the planes. Angles are frequently denoted by lower-case Greek letters.
Cf. acute, obtuse, right angle.
anxiety, math
A fear or emotional dislike of mathematics, characterized by difficulty learning or mastering mathematical techniques or concepts, poor performance on exams despite careful preparation, etc.
Related MiniText: Coping With Math Anxiety
arc
Graph Theory: Another name for a directed edge.
arithmetic
The mathematical theory of addition, subtraction, multiplication, and division of integers. Formally, arithmetic is usually axiomatized by the so-called Peano axioms, and the theory is then often referred to as PA (for “Peano arithmetic”).
associative
An operation “ · ” on a set A is associative if for all a, b, and c in A, (a · b) · c = a · (b · c).
Cf. commutative, distributive.
associative property
A property of numbers which states that the operations of addition and multiplication are associative.
Cf. commutative, distributive.
axiom
In formal mathematics, a formula or schema of formulas stipulated as true in the theory under discussion, i.e., assumed to be true at the outset, and so not requiring proof. Axioms are the counterpart in mathematics of suppositions, assumptions, or premises in ordinary syllogistic logic.
base
Number systems: In a place-notational number system, the number of symbols used. For example, in base two the two symbols 0 and 1 are used, and in base seven the seven symbols 0, 1, 2, 3, 4, 5, and 6 are used.
Exponential expressions: The number or expression which is “being raised to” the power of the exponent.
Topology: Given a topological space X, a base is a class B of sets such that, for every x in X and every neighborhood U of x, there is a set b in B such that x is contained in b and b is contained in U.
bijection
A bijective function, i.e., a function that is both an injection and a surjection.
bijective
A function is bijective if it is both injective and surjective, i.e., both “one-to-one” and “onto.”
binary operation
A binary operation on a set X is a function whose domain is the set of ordered pairs of elements from X and whose range is X.
Examples: addition on the set of integers, function composition.
bound
A lower bound of any subset of a linear order (linearly ordered set) is an element which is less than or equal to every element of the subset. The greatest lower bound is the largest of its lower bounds. An upper bound of any subset of a linear order is an element which is greater than or equal to every element of the subset. The least upper bound is the smallest of the upper bounds.
Cf. infimum, supremum.
bounded
A set or sequence of values is called bounded if there is a value M such that the values are never bigger than M and never smaller than –M. A function is bounded if its values are bounded. A sequence of functions is pointwise bounded if at every domain element x the sequence of values of the functions at x is bounded. A subset E of a locally compact topological space is bounded if there exists a compact set C such that E is contained in C. Such a subset is called s-bounded if there exists a sequence of compact sets Ci such that E is contained in their union. See also: totally bounded.
Cartesian plane
The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.
Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane.
chord
A straight line segment connecting two points on a curve or surface.
circle
In a plane, the locus of all points equidistant from a given point, called the center. The general equation for a circle in the Cartesian plane is given by (x - h) 2 + (y - k) 2 = r 2, where r is the radius of the circle (distance from the center to the locus of points), and (h, k) are the coordinates of the center.
The interior of a circle is referred to as an open disk.
A circle is also a conic section; a special case of an ellipse in which the foci coincide.
Related article: Conics
circumference
Geometry: The distance around a circle in the plane, or around a great circle of a sphere.
Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles.
closed
General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.
Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.
See the following listings for other uses of the word “closed” in mathematics.
Topology: A set is topologically closed if it is not open.
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