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algebra – axiom
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 lattice
A lattice L is said to be algebraic if every element of L is a join of compact elements of L.
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.
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.
almost disjoint
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.
Cf. D-system.
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.
antichain
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 .
Cf. chain.
antinomy
A paradox arising from formal mathematical axioms. An antinomy may lead to a strict contradiction, as in the Russell Paradox, or merely offend our intuition, as in the Banach-Tarski Paradox.
antisymmetric relation
A relation “~” on a set X is antisymmetric if, for all x and y in X, whenever x ~ y and y ~ x then x = y.
Cf. symmetric relation, asymmetric relation.
antitone function
A function that is order preserving and decreasing.
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”).
Aronszajn tree
Set Theory: For a a regular cardinal, an a-Aronszajn tree is a tree T with |T| = a, such that every chain in T is of cardinality less than a. Every Suslin tree is an Aronszajn tree. There are no Aronszajn trees of height w.
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.
asymmetric relation
A relation “~” on a set X is asymmetric if, for all x and y in X, x ~ y implies that it is not the case that y ~ x.
Cf. symmetric relation, antisymmetric relation.
automorphism
An isomorphism of a space to itself, that is, a bijective function on a space that preserves relationships, operations, and/or all essential properties (continuity, order, etc.).
Cf. homomorphism, group automorphism, ring automorphism, field automorphism.
automorphism group
The automorphisms of any object or structure can be composed to produce new automorphisms of that object. Since functional composition is associative, and since automorphisms are by their nature invertible, the automorphisms of an object or structure form a group (with the trivial automorphism serving as the identity element). This group is called the automorphism group of that object, usually denoted by Aut(X) for some object X. Automorphism groups are very useful, as they yield information about the symmetries of objects at hand.
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.
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