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A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ). In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.
If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A. Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for proper subsets. For clarity, one can ...
Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal.
For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
The set of all dense open subsets of a (non–empty) topological space is a proper π –system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π –system and a prefilter that is finer than .
An ideal in the sense of ring theory, usually of a Boolean algebra, especially the Boolean algebra of subsets of a set iff if and only if improper See proper, below. inaccessible cardinal A (weakly or strongly) inaccessible cardinal is a regular uncountable cardinal that is a (weak or strong) limit indecomposable ordinal
If the base set is finite, then = ℘ since every subset of , and in particular every complement, is then finite.This case is sometimes excluded by definition or else called the improper filter on . [2] Allowing to be finite creates a single exception to the Fréchet filter’s being free and non-principal since a filter on a finite set cannot be free and a non-principal filter cannot contain ...
Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition. Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H.