Function application explained

In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction.

Representation

Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in parentheses. For example, the following expression represents the application of the function ƒ to its argument x.

f(x)

In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just by juxtaposition. For example, the following expression can be considered the same as the previous one:

f x

The latter notation is especially useful in combination with the currying isomorphism. Given a function

f:(X x Y)\toZ

, its application is represented as

f(x,y)

by the former notation and

f (x,y)

(or

f \langlex,y\rangle

with the argument

\langlex,y\rangle\inX x Y

written with the less common angle brackets) by the latter. However, functions in curried form

f:X\to(Y\toZ)

can be represented by juxtaposing their arguments:

f x y

, rather than

f(x)(y)

. This relies on function application being left-associative.

When mathematical notation is represented in a digital document, the invisible zero-width Unicode characters and can be used to distinguish concatenation meaning function application from concatenation meaning multiplication

Set theory

In axiomatic set theory, especially Zermelo–Fraenkel set theory, a function

f:D\mapstoR

is often defined as a relation (

f\subseteqD x R

) having the property that, for any

x\inD

there is a unique

y\inR

such that

(x,y)\inf

One is usually not content to write "

(x,y)\inf

" to specify that

, and usually wishes for the more common function notation "

f(x)=y

", thus function application, or more specifically, the notation "

f(x)

", is allowed by defining a binary function symbol

\bullet(\bullet)

as:^[1]

$X(Y) = \left\$

Notes and References

Book: Mendelson, Elliot . Elliott Mendelson . Introduction to Mathematical Logic . 1987 . . 978-1-4615-7290-9 . 3rd . Belmont . 185 . 10.1007/978-1-4615-7288-6 . 86-11084.