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If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
The complement of A is the set of all elements (of U) that do not belong to A. It may be denoted A c or A′. In set-builder notation, = {:}. The complement may also be called the absolute complement to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the ...
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".
The complement of a closed nowhere dense set is a dense open set. Given a topological space X , {\displaystyle X,} a subset A {\displaystyle A} of X {\displaystyle X} that can be expressed as the union of countably many nowhere dense subsets of X {\displaystyle X} is called meagre .
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. [1] [2] In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.
The complement of a convex set, such as the epigraph of a concave function, ... As the definition of a convex set is the case r = 2, ...
The cofinite topology (sometimes called the finite complement topology) is a topology that can be defined on every set . It has precisely the empty set and all cofinite subsets of X {\displaystyle X} as open sets.
If : ℘ [,] is an outer measure on a set , where (by definition) the domain is necessarily the power set ℘ of , then a subset is called –measurable or Carathéodory-measurable if it satisfies the following Carathéodory's criterion: = + (), where := is the complement of .