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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 ...
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
In mathematics, a field of sets is a mathematical structure consisting of a pair (,) consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in , finite unions, and finite intersections.
The set E of all finite definitions of real numbers is a subset of A. As A is countable, so is E. Let p be the nth decimal of the nth real number defined by the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal if p is not equal either to 8 or 9, and unity if p is equal to 8 or 9.
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection of subsets of a given set is called a family of subsets of , or a family of sets over .
A subset V of E is a subanalytic set if for each point there exists a relatively compact semianalytic set X in a Euclidean space F of dimension at least as great as E, and a neighbourhood U in E, such that the intersection of V and U is a linear projection of X into E from F. In particular all semianalytic sets are subanalytic.
In Mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
Cantor's paradox: The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.