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Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set". [ 1 ] However, null set is a distinct notion within the context of measure theory , in which it describes a set of measure zero (which is not necessarily empty).
For any non-empty set X, P = { X} is a partition of X, called the trivial partition. Particularly, every singleton set {x} has exactly one partition, namely { {x} }. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely, { A, U ∖ A}.
Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space = (,,) a null set is a set such that () =
A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). [6]
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 ).
Let S be a non-empty set of real numbers.. A real number x is called an upper bound for S if x ≥ s for all s ∈ S.; A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S.
For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B. The negation of the axiom can thus be expressed as: There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.
Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. [2] For instance, the three sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not disjoint. In fact, there are no two disjoint ...