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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 () =
The set {} is empty and thus not inhabited. Naturally, the example section thus focuses on non-empty sets that are not provably inhabited. It is easy to give such examples by using the axiom of separation, as with it logical statements can always be
However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice. For example, V ω ∖ { ∅ } {\displaystyle V_{\omega }\setminus \{\emptyset \}} has a choice function, where V ω {\displaystyle V_{\omega }} is the set of hereditarily finite sets , i.e. the first set ...
Every non-empty subset of the real numbers which is bounded from above has a least upper bound.. In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers.
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.
Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. However, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a ...