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The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets.
The notation for this last concept can vary considerably. Set theorists will sometimes write " ... Algebra of sets – Identities and relationships involving sets;
Algebra of sets – Identities and relationships involving sets; Alternation (formal language theory) – in formal language theory and pattern matching, the union of two sets of strings or patterns − the union of sets of strings; Axiom of union – Concept in axiomatic set theory
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers.
Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets Generalized quantifier – Expression denoting a set of sets in formal semantics Indexed family – Collection of objects, each associated with an element from some index set
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