Empty function

Encyclopedia

In mathematics

, an

whose domain is the empty set

. For each set

The graph

of an empty function is a subset

of the Cartesian product

∅×

since there

Most authors will not care, when defining the term “constant function

” precisely, whether or not the empty function qualifies, and will use whatever definition is most convenient. Sometimes, however, it is best not to consider the empty function to be constant, and a definition that makes reference to the range is preferable in those situations. This is much along the same lines of not considering 1 to be a prime number

, an empty topological space

to be connected, or the trivial group

to be simple

.

The existence of an empty function from ∅ to ∅ is required to make the category of sets

a category. (In a category, each object needs to have an "identity morphism", and only the empty function is the identity on the object ∅.)

The existence of a unique empty function from ∅ into each set

in the category of sets. In terms of cardinal arithmetic, it means that

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an

**empty function**is a functionFunction (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

whose domain is the empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

. For each set

*A*, there is exactly one such empty functionThe graph

Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

of an empty function is a subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

of the Cartesian product

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

∅×

*A*. Since the product is empty the only such subset is the empty set ∅. The empty subset is a valid graph since for every*x*in the domain ∅ there is a unique*y*in the codomain*A*such that (*x*,*y*) ∈ ∅. This is an example of a vacuous truthVacuous truth

A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...

since there

*is no x in the domain*.Most authors will not care, when defining the term “constant function

Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...

” precisely, whether or not the empty function qualifies, and will use whatever definition is most convenient. Sometimes, however, it is best not to consider the empty function to be constant, and a definition that makes reference to the range is preferable in those situations. This is much along the same lines of not considering 1 to be a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, an empty topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

to be connected, or the trivial group

Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...

to be simple

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

.

The existence of an empty function from ∅ to ∅ is required to make the category of sets

Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

a category. (In a category, each object needs to have an "identity morphism", and only the empty function is the identity on the object ∅.)

The existence of a unique empty function from ∅ into each set

*A*means that the empty set is an initial objectInitial object

In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...

in the category of sets. In terms of cardinal arithmetic, it means that

*k*^{0}= 1 for every cardinal numberCardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

*k*- particularly profound when*k=0*to illustrate the strong statement of indices pertaining to*0*.