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It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
The great variety and (relative) complexity of formulas involving set subtraction (compared to those without it) is in part due to the fact that unlike ,, and , set subtraction is neither associative nor commutative and it also is not left distributive over ,, , or even over itself.
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. [1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero ( ) sets and it is by definition equal to the empty set.
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅)
The axiom of extensionality, [1] [2] also called the axiom of extent, [3] [4] is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. [5] [6] The axiom defines what a set is. [1] Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.
In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and set membership predicates, respectively. The first level of the Lévy hierarchy is defined as containing only formulas with no unbounded quantifiers and is denoted by Δ 0 = Σ 0 = Π 0 {\displaystyle \Delta _{0}=\Sigma _{0}=\Pi _{0}} . [ 1 ]
A formula of set theory is stratified if and only if there is a function which sends each variable appearing in (considered as an item of syntax) to a natural number (this works equally well if all integers are used) in such a way that any atomic formula appearing in satisfies () + = and any atomic formula = appearing in satisfies () = ().