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almost disjoint – bijection

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.

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 XN, 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   A test for the convergence of a series. See the article for a complete description.

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.

 Archimedes
Archimedes
(born 287 bce)   Greek geometer and applied mathematician who lived at Syracuse, a Greek settlement on the coast of Sicily. He is famous both for his profound mathematical work and for his ingenious mechanical inventions, which at the end of his life he used to defend his city against the Romans during the Second Punic War. When he discovered the hydrostatic principle – that a floating body of a given weight displaces a volume of liquid having an equal weight – he is reputed to have jumped from his bath and run naked through the streets shouting “Eureka!” (I have found it!) Archimedes is credited with many brilliant mathematical discoveries, especially in properties of geometrical figures, areas and volumes of conics, and principles of trigonometry. (He is thought to have known Heron's formula – several centuries before Heron.) He found areas of plane figures by the method of exhaustion, prefiguring modern integral calculus. He was killed by a Roman soldier during the sack of Syracuse.

 Aristotle
Aristotle
(born 384 bce)   Greek philosopher and scientist who developed syllogistic logic as a formal scholastic discipline. He defined the syllogism as a “discourse in which certain things having been stated, something else follows of necessity from their being so.” Aristotle’s philosophy of the infinite, in which he held that only the potential infinite, and not any collection which was actually infinite, could be entertained by the reason, held great authority for 2,000 years, until Georg Cantor introduced his theory of transfinite sets in the 19th century. Aristotle wrote widely on every field of learning, interpreted and disagreed with the writings of his teacher Plato, and served as tutor to Alexander the Great.

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.

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.

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.

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

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.

bijection   A bijective function, i.e., a function that is both an injection and a surjection.

almost disjoint – bijection

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