Information about Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.
Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets).
A class that is not a set is called a proper class.
Many objects in mathematics are too big for sets and need to be described with classes. Examples include large categories or the class-field of surreal numbers. The usual construction to show that a given "thing" is a proper class is to show that such a "thing" has at least as many elements as there are ordinal numbers. For an example of such a proof, see free lattices.
A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. Instead, these paradoxes become proofs that a certain class is proper. For example, Russell's paradox becomes a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox becomes a proof that the class of all ordinal numbers is proper.
The standard Zermelo-Fraenkel set theory axioms do not talk about classes; classes exist only in the metalanguage as equivalence classes of logical formulas. Another approach is taken by the von Neumann-Bernays-Gödel axioms; classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. The proper classes, then, are those classes that are not elements of any other class.
In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all collections are sets) but the criterion of sethood is not size. For example, any set theory with a universal set has proper classes which are subclasses of sets.
The word "class" is sometimes used synonymously with "set," most notably in the term "equivalence class." This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
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Many objects in mathematics are too big for sets and need to be described with classes. Examples include large categories or the class-field of surreal numbers. The usual construction to show that a given "thing" is a proper class is to show that such a "thing" has at least as many elements as there are ordinal numbers. For an example of such a proof, see free lattices.
A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. Instead, these paradoxes become proofs that a certain class is proper. For example, Russell's paradox becomes a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox becomes a proof that the class of all ordinal numbers is proper.
The standard Zermelo-Fraenkel set theory axioms do not talk about classes; classes exist only in the metalanguage as equivalence classes of logical formulas. Another approach is taken by the von Neumann-Bernays-Gödel axioms; classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. The proper classes, then, are those classes that are not elements of any other class.
In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all collections are sets) but the criterion of sethood is not size. For example, any set theory with a universal set has proper classes which are subclasses of sets.
The word "class" is sometimes used synonymously with "set," most notably in the term "equivalence class." This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, often as Venn diagrams, of collections of objects, and the elements of, and membership in, such collections.
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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SET may stand for:
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- Sanlih Entertainment Television, a television channel in Taiwan
- Secure electronic transaction, a protocol used for credit card processing,
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The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
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In mathematics, the parity of an object refers to whether it is even or odd.
The formal definition of an odd number is an integer of the form n=2k +1, where k is an integer. The definition of an even number is n=2k where k is an integer.
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The formal definition of an odd number is an integer of the form n=2k +1, where k is an integer. The definition of an even number is n=2k where k is an integer.
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ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them.
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In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
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In mathematics, surreal numbers are the elements of a field[1] containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar
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In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property. The word problem for free lattices is also challenging.
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Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, often as Venn diagrams, of collections of objects, and the elements of, and membership in, such collections.
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naive set theory[1] is one. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of
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In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics.
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Part of the foundation of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.
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In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.
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ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them.
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metalanguage is a language used to make statements about other languages (object languages). Formal syntactic models for the description of grammar, e.g. generative grammar, are a type of metalanguage.
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In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica.
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In set theory, a semiset is a proper class which is contained in a set. The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek.
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equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:
The notion of equivalence classes is useful for constructing sets out of already constructed ones.
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- [a] =
The notion of equivalence classes is useful for constructing sets out of already constructed ones.
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For the periodical, see .
The 19th Century (also written XIX century) lasted from 1801 through 1900 in the Gregorian calendar. It is often referred to as the "1800s...... Read more.