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base – Carmichael number
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.
Bolzano-Weierstrass property
A topological space X is said to have the Bolzano-Weierstrass property if every infinite sequence in X has a convergent subsequence. It is a theorem that every compact space has the Bolzano-Weierstrass property.
Borel measure
A measure m on a metric space X defined on a s-algebra of sets which contains the Borel sets is called a Borel measure. See also: Lebesgue measure.
Borel set
If X is a topological space, then the s-algebra of sets generated by the open sets of X is called the Borel s-algebra, and its members are called Borel sets. The Borel subsets of the real line are generated by the open intervals (equivalently, by the closed intervals, half-open intervals, and rays).
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.
bounded variation
A function on the real numbers with range in the complex numbers is said to be of bounded variation if its total variation is finite. The space of all such functions is denoted by BV. If in addition f is right continuous and its limit at negative infinity is zero, then f is said to be of normalized bounded variation. The space of such functions is denoted by NBV.
Bourbaki, Nicolas
Although the myth of an actual French mathematician named Bourbaki was maintained for many years, it became generally known in 1952 that “Bourbaki” is the nom de plume of a private congress of young French mathematicians which formed in 1935, with the purpose of writing a comprehensive exposition of modern mathematics in many volumes, called Elements. Bourbaki’s original members included André Weil, Henri Cartan, Claude Chevalley, and Jean Dieudonné. Other mathematicians have been recruited into Bourbaki in the years since, including G. de Rham, A. Grothendieck, S. Lang, and R. Thom. A principle motivation for the activity of the Bourbaki group is the felt need to unify the entire corpus of mathematical work in the face of accelerating diversification and specialization of mathematics.
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Georg Cantor
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Cantor, Georg
(born 1845)
German mathematician and founder of modern set theory. Cantor’s concern with the set-theoretic nature of the real numbers began while he was working on certain properties of trigonometric series during the early 1870’s. Inspired by the work of his friend Dedekind, Cantor proved in 1873 that the rational numbers are countable, and later the same year proved that the real numbers themselves are uncountable. These proofs introduced important methods, including the notion of a one-to-one correspondence and the method of diagonalization. Cantor’s results were counterintuitive and broke with established dogma about the nature of infinity. Consequently, he became a controversial figure, opposed by some mathematicians (e.g., Kronecker) who used their influence to stifle Cantor’s career as much as possible. This opposition exacerbated Cantor’s predisposition to severe depression, and he ultimately died, of a heart attack, in a sanitorium.
Related article: Cantor's Theorem
Related article: Cantor Set
Related MiniText: Infinity -- You Can't Get There From Here...
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...
Carathéodory’s Theorem
See: outer measure.
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
Carmichael number
A composite integer n which satisfies the congruence an = a modulo n for all integers a.
Thus they pass the Fermat test without being prime.
Carmichael numbers are characterised by the Korselt condition that they are composite, squarefree and p-1 | n-1 for every prime factor p | n.
The first Carmichael number is 561 = 3 × 11 × 17.
It was proved in 1994 by Alford, Granville and Pomerance that there are infinitely many Carmichael numbers.
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