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antichain – bijective

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.

 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.

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.

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.

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.

bijective   A function is bijective if it is both injective and surjective, i.e., both “one-to-one” and “onto.”

antichain – bijective

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