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The empty set is the set containing no elements. In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in
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 ).
a pre-measure if it is non-negative, countably additive (including finitely additive), and has a null empty set. a measure if it is a pre-measure whose domain is a σ-algebra. That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a null empty set.
In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions ...
For every set X, there is a unique function, called the empty function, or empty map, from the empty set to X. The graph of an empty function is the empty set. [note 5] The existence of empty functions
The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.
The kernel of the empty set, , is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty. [3] A family is said to be free if it is not fixed; that is, if its kernel is the empty set. [3]
[3] [5] In measure theory, a null set is a (possibly nonempty) set with zero measure. 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 ...