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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
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
In set theory, the empty set, that is, the set with zero elements, denoted "{}" or "∅", may also be called null set. [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).
The empty set is the unique initial object in Set, the category of sets. Every one-element set is a terminal object in this category; there are no zero objects. 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.
Note that a null set is not necessarily an empty set. Common notations for the empty set include "{}", "∅", and "". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets (and not related in any way to the Greek letter Φ). [2] Empty sets ...
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 ).
In mathematics, an algebraic structure or algebraic system [1] consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.