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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).
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 () =
In axiomatic set theory, the axiom of empty set, [1] [2] also called the axiom of null set [3] and the axiom of existence, [4] [5] is a statement that asserts the existence of a set with no elements. [3]
A null space of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element). For example, in linear algebra, the null space of a linear mapping, also known as kernel, is the set of vectors which map to the null vector under that mapping. In statistics, a null hypothesis is a ...
The null sign (∅) is often used in mathematics for denoting the empty set. The same letter in linguistics represents zero , the lack of an element. It is commonly used in phonology , morphology , and syntax .
Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. In the category Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object. Morphisms of pointed sets.
The empty set is also occasionally called the null set, [11] though this name is ambiguous and can lead to several interpretations. The power set of a set A, denoted (), is the set whose members are all of the possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }. Notably, () contains both A and the empty set.
The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "it is an empty set". It is like the difference between saying "There is no bucket" and saying "The ...