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Naive set theory | A Wisdom Archive on Naive set theory | | Naive set theory A selection of articles related to Naive set theory | |
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Naive set theory
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ARTICLES RELATED TO Naive set theory | | | |  Naive set theory: Encyclopedia II - Naive set theory - Unions intersections and relative complementsGiven two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B.
The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B.
Finally, the relative complement of B relative to A, also known as the set theoretic differenc ...
See also:Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Unions intersections and relative complements |
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| | | | Naive set theory: Encyclopedia II - Complement set theory - Relative complement If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A.
The relative complement of A in B is usually written B − A (also B \ A).
Formally:
Examples:
{1,2,3} − {2,3,4} = {1}
{2,3,4} −& ...
See also:Complement set theory, Complement set theory - Relative complement, Complement set theory - Absolute complement Read more here: » Complement set theory: Encyclopedia II - Complement set theory - Relative complement |
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| | | | Naive set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory |
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| | | | Naive set theory: Encyclopedia II - Union set theory - Algebraic properties Binary union (the union of just two sets at a time) is an associative operation; that is,
A ∪(B ∪ C) = (A ∪ B) ∪ C.
In fact, A ∪ B ∪ C is equal to both of these sets as well, so parentheses are never needed when writing only unions. Similarly, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union. That is, {} ∪ A = A, for any set A. Thus one can t ...
See also:Union set theory, Union set theory - Basic definition, Union set theory - Finite unions, Union set theory - Algebraic properties, Union set theory - Infinite unions Read more here: » Union set theory: Encyclopedia II - Union set theory - Algebraic properties |
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| | | | Naive set theory: Encyclopedia II - Axiomatic set theory - Independence in ZFC Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Independence in ZFC |
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| | | | Naive set theory: Encyclopedia - Mathematics Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...
Including:
Read more here: » Mathematics: Encyclopedia - Mathematics |
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| | | | Naive set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory |
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| | | | Naive set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC.
An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory |
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| | | | Naive set theory: Encyclopedia II - Union set theory - Finite unions More generally, one can take the union of several sets at once. The union of A, B, and C, for example, contains all elements of A, all elements of B, and all elements of C, and nothing else. Formally, x is an element of A ∪ B ∪ C if x is in A or x is in B or x is in C.
Union is an associative operation, it doesn't matter in what order unions are taken. In mathematics a finite union means any union carried out on a finite number of sets: it ...
See also:Union set theory, Union set theory - Basic definition, Union set theory - Finite unions, Union set theory - Algebraic properties, Union set theory - Infinite unions Read more here: » Union set theory: Encyclopedia II - Union set theory - Finite unions |
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| | | | Naive set theory: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness:
a set, x0, is well founded iff it has no infinite descending membership sequence:
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In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets |
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| | | | Naive set theory: Encyclopedia II - Intersection set theory - Basic definition The intersection of A and B is written "A ∩ B". Formally:
x is an element of A ∩ B if and only if
x is an element of A and
x is an element of B.
For example, the intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. The number 9 is not contained in the intersection of the set of prime numbers
{2, 3, 5, 7, 11, … ...
See also:Intersection set theory, Intersection set theory - Basic definition, Intersection set theory - Arbitrary intersections, Intersection set theory - Nullary intersection Read more here: » Intersection set theory: Encyclopedia II - Intersection set theory - Basic definition |
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| | | | Naive set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function , etc.
For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify th ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics |
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