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In set theory, the complement of a set A, often denoted by (or A′ ), [1] is the set of elements not in A. [2] When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A .
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.
In mathematics, a relation on a set may, or may not, hold between two given members of the set. 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 values 3 and 1 nor between 4 and 4 , that is, 3 < 1 and 4 < 4 both evaluate to ...
The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes. [4] More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates). [5] The elements of a set may be manifested in music as simultaneous chords, successive tones (as in ...
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom , which states that for each set there is a Grothendieck universe it belongs to (see below).
Descriptive set theory begins with the study of Polish spaces and their Borel sets . A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line , the Baire space , the Cantor space , and ...
Categorical set theory is any one of several versions of set theory developed from or treated in the context of mathematical category theory. See also ...
