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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.
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.
Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can ...
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.
Ordinal addition on the natural numbers is the same as standard addition. The first transfinite ordinal is ω , the set of all natural numbers, followed by ω + 1 , ω + 2 , etc. The ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first.
The kernel of the sunflower is the brown part in the middle, and each set of the sunflower is the union of a petal and the kernel. In the mathematical fields of set theory and extremal combinatorics, a sunflower or -system [1] is a collection of sets in which all possible distinct pairs of sets share the same intersection.