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automorphism – Cartesian product
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.”
binary operation
A binary operation on a set X is a function whose domain is the set of ordered pairs of elements from X and whose range is X.
Examples: addition on the set of integers, function composition.
bound
A lower bound of any subset of a linear order (linearly ordered set) is an element which is less than or equal to every element of the subset. The greatest lower bound is the largest of its lower bounds. An upper bound of any subset of a linear order is an element which is greater than or equal to every element of the subset. The least upper bound is the smallest of the upper bounds.
Cf. infimum, supremum.
bounded
A set or sequence of values is called bounded if there is a value M such that the values are never bigger than M and never smaller than –M. A function is bounded if its values are bounded. A sequence of functions is pointwise bounded if at every domain element x the sequence of values of the functions at x is bounded. A subset E of a locally compact topological space is bounded if there exists a compact set C such that E is contained in C. Such a subset is called s-bounded if there exists a sequence of compact sets Ci such that E is contained in their union. See also: totally bounded.
Cantor-Bendixson Theorem
Every closed subset of the real numbers is the disjoint union of a perfect set and a countable set.
Cf. derived set.
Cantor-Schröder-Bernstein Theorem
For any sets A and B, if there exists an injective (one-to-one) function from A into B and also an injective function from B into A, then there exists a surjective (onto) function from A onto B.
Cantor set
ARTICLE
A subset of the unit interval (on the real number line) which is a perfect set, nowhere dense, uncountable, and homeomorphic to the entire unit interval. See the article for a complete exposition.
Cantor’s Theorem
ARTICLE
Given any set, a new set may be constructed which is cardinally larger, i.e., whose size is represented by a larger cardinal than the size of the initial set. More specifically, for no set X is there a bijection of X onto its power set. This theorem establishes both that there are different sizes of infinity, and that there is no greatest such “size.” Cantor’s proof was the first use of a diagonalization argument to derive a contradiction.
Related MiniText: Infinity -- You Can't Get There From Here...
cardinal
A cardinal number represents the size of a set irrespective of the order or structure of its elements. If X is a set, its cardinality is generally indicated by |X|. Two sets A and B are said to have the same cardinality if there is a function from A into B that is bijective (i.e., one-to-one and onto).
A cardinal is an initial ordinal, i.e., an ordinal for which there does not exist a bijection onto any lesser ordinal. Thus, all finite ordinals (i.e., the natural numbers) are cardinals, but most transfinite ordinals are not cardinals.
If a is a cardinal, then we denote by a+ the least cardinal greater than a. A cardinal k is called a successor cardinal if k = a+ for some cardinal a. Otherwise k is called a limit cardinal. The heirarchy of transfinite cardinals is defined by recursion, and denoted using the Hebrew letter aleph:
where w is the first infinite ordinal.
Cf. cardinal arithmetic, inaccessible cardinal, regular.
Related MiniText: Infinity -- You Can't Get There From Here...
cardinal arithmetic
Let k and l denote cardinals and A and B denote sets such that k = |A| and l = |B|. Denote by BA the set of all functions from B to A, and by A × B the Cartesian product of A and B. The operations of cardinal arithmetic are then defined by- k + l = |A union B|;
- k • l = |A × B|;
- kl = |BA|
For finite cardinals, this corresponds to the operations of ordinary arithmetic, but is quite different in the case of transfinite cardinals. For instance, if k is not finite, then k + k = k.
Cf. ordinal arithmetic.
cardinality
Two sets X and Y have the same cardinality if there exists a bijection between them. Specifically, the cardinality of X may be understood as a canonical object meant to represent the proper class of sets to which X is bijective. If the axiom of choice is assumed, then the cardinality of X is the unique cardinal number having the same cardinality as X
Carroll’s Paradox
ARTICLE
Paradox published by Lewis Carroll as “What the Tortoise Said to Achilles.” See the article for a full exposition.
Cf. Charles Lutwidge Dodgson.
Cartesian plane
The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.
Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane.
Cartesian product
For any collection {Ai}, i = 1, 2, 3, ..., n, of sets, the Cartesian product
is the set of ordered n-tuples (a1,a2, ... ,an) with a1 an element of A1, a2 an element of A2, etc. The Cartesian product R2 of the set of real numbers is called the Cartesian plane, and in general n-dimensional real space is the Cartesian product Rn. The assertion that the Cartesian product of an infinite collection of non-empty sets is non-empty is equivalent to the axiom of choice.
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