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These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line {}, which consists of the real numbers and .
The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, [1] and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of ...
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
As an illustration, let R be the set of real numbers, let Z be the set of integers, let O be the set of odd integers, and let P be the set of current or former U.S. Presidents. Then O is a subset of Z, Z is a subset of R, and (hence) O is a subset of R, where in all cases subset may even be read as proper subset. Not all sets are comparable in ...
A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but also one of the set S as subset of P. A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest ...
Subspace, a particular subset of a parent space; A subset of a topological space endowed with the subspace topology; Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.