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algebraic function – bijection
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.
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:- fg is in A,
- f + g is in A, and
- for all constants c, cf is in A
whenever f and g are 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.
almost everywhere
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."
alternating series test
ARTICLE
A test for the convergence of a series. See the article for a complete description.
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.
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”).
Ascoli-Arzelŕ Theorem
If {X} is a compact Hausdorff space and F is an equicontinuous, pointwise bounded subset of the space C(X) of continuous functions on X, then F is totally bounded in the uniform metric and the closure of F in C(X) is compact. Also, if X is a s-compact, locally compact Hausdorff space and if (fn} is an equicontinuous, pointwise bounded sequence in C(X), then there exists a subsequence and an f in C(X) such that the subsequence converges to f uniformly on compact sets.
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.
axiom of choice
An axiom of set theory, particularly ZFC, which asserts that from any (infinite) family of sets a new set may be created containing exactly one element from each set in the family. This axiom has been shown to be independent of the other axioms of ZFC (ZF). It is equivalent to: 1) Zorn's Lemma; 2) the Hausdorff Maximality Theorem; 3) the well-ordering principle; and 4) the statement that the cartesian product of an infinite family of sets is non-empty. Because the axiom of choice permits non-constructive proofs, it is rejected by intuitionism, and careful mathematicians refer explicitly to its use in proofs which require it.
Baire Category Theorem
If X is a complete metric space, then the intersection of a countable collection of dense open subsets of X is also dense. Equivalently, the union of a countable collection of closed, nowhere dense subsets of X is nowhere dense.
Banach space
A complete normed space.
Banach-Tarski Paradox
Given two bounded subsets of 3-dimensional Euclidean space, provided each has interior points, then the first may be decomposed into finitely many pieces and translated by rigid motions into a subset congruent to the second subset. For instance, a ball of unit radius can be decomposed into finitely many pieces, and then reassembled into two balls of unit radius. This shocking result is a consequence of the axiom of choice.
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.
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