accumulation point antichain
Given a set X, an accumulation point of X is a point p (which may or may not be in X) such that every neighborhood of p contains points of X different from p. See also: limit point, perfect set.
Abbreviation of arc-cosine.
An infinite set considered as a completed whole, e.g., the natural numbers in modern set theory. Aristotle distinguished the actually infinite from the potential infinite, and considered that only the latter was permissible to thought.
An angle is called acute if it is less than a right angle, that is, if its measure is less than 90° (p/2 radians). A triangle is called acute if all of its angles are acute.
To perform the operation of addition on numbers or quantities to obtain a total value. See addition.
A binary operation on numbers or quantities to obtain a total value, called the sum. For natural numbers this operation is defined recursively by the Peano axioms (i) a + 0 = a, and (ii) a + Sb = S(a + b), where Sa is the successor of a. For sets, addition is defined by the cardinality of the disjoint union. Addition on other kinds of numbers or structures is typically defined as an extension of these operations.
Two vertices of a graph are said to be adjacent when they have an edge that is incident on both of them. Also occasionally used of edges that are incident on at least one vertex in common.
See: almost everywhere.
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."
A field F is algebraically closed if every polynomial with coefficients in F has a root in F. Neither the field of rational numbers nor the field of real numbers is algebraically closed, but the field of complex numbers is algebraically closed.
A function which operates on its variable(s) by addition, subtraction, multiplication, division, raising to a power, or extraction of roots. Compare: transcendental function.
A lattice L is said to be algebraic if every element of L is a join of compact elements of L.
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.
algebra of functions
Given a topological space X, an algebra of functions on X is a subset A of the space of all continuous functions on X such that for all functions f and g in X we have:
whenever f and g are in A.
- fg is in A,
- f + g is in A, and
- for all constants c, cf is in A
algebra of sets
Given a set X, an algebra on X is a collection of subsets of X which is closed under finite unions and complements. Such an algebra always includes the empty set and X itself. An algebra may also be defined as a ring of sets which includes X. An algebra of sets which is closed under countable unions is called a s-algebra.
Set Theory: If a is an infinite cardinal and x and y are subsets of a, then x and y are said to be almost disjoint of their intersection has cardinality less than a. An almost disjoint family is a subset A of the power set of a, such that any two distinct members of A of size a are almost disjoint. A maximal almost disjoint family is an almost disjoint family not properly contained in any other almost disjoint family. It is a theorem that if A is an almost disjoint family and |A| = a, then A is not maximal.
Given a measure m on a measure space X, a condition (e.g., continuity of functions, etc.) is said to hold almost everwhere if it holds on X – N, i.e., on the set difference of X and N, where N is a null set. This is often abbreviated by "a.e."
The subgroup of a symmetric group on n elements which contains all of the even permutations is called the alternating group on n elements.
alternating series test
A test for the convergence of a series. See the article for a complete description.
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.
If X is a set with a strict partial order , then a subset A of X is called an antichain if no x and y in A are related by .