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In the left hand sides of the following identities, is the L eft most set and is the R ight most set. Whenever necessary, both L and R {\displaystyle L{\text{ and }}R} should be assumed to be subsets of some universe set X , {\displaystyle X,} so that L ∁ := X ∖ L and R ∁ := X ∖ R . {\displaystyle L^{\complement }:=X\setminus L{\text ...
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".
In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...
In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a ...
Algebra of sets – Identities and relationships involving sets; Class (set theory) – Collection of sets in mathematics that can be defined based on a property of its members; Combinatorial design – Symmetric arrangement of finite sets; δ-ring – Ring closed under countable intersections
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The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.