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# OpenMath Content Dictionary: fns1

Canonical URL:
http://www.openmath.org/cd/fns1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
fns1.ocd
CD as XML Encoded OpenMath:
fns1.omcd
Defines:
domain, domainofapplication, identity, image, inverse, lambda, left_compose, left_inverse, range, restriction, right_inverse
Date:
2009-04-01
Version:
4
Review Date:
2014-04-01
Status:
official

```
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work.  Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society.  In particular, if there is a Content
Dictionary Group whose name is, for example, `math' containing
Content Dictionaries named `math1', `math2' etc., then you should
not name a derived Content Dictionary `mathN' where N is an integer.
However you are free to name it `private_mathN' or some such.  This
is because the names `mathN' may be used by the OpenMath Society
for future extensions.
compilation of derived works, but keep paragraphs a) and b)
intact.  The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
society at http://www.openmath.org.
```

This CD is intended to be `compatible' with the corresponding elements in Content MathML.

In this CD we give a set of functions concerning functions themselves. Functions can be constructed from expressions via a lambda expression. Also there are basic function functions like compose, etc.

## restriction

Role:
application
Description:

restriction takes two arguments, a function f, and a set S, which should be a subset of domain(f) and returns the function f restricted to S.

Formal Mathematical property (FMP):
$\mathrm{domain}\left({f|}_{S}\right)=S$
Formal Mathematical property (FMP):
${f|}_{\mathrm{domain}\left(f\right)}=f$
Signatures:
sts

 [Next: domainofapplication] [Last: lambda] [Top]

## domainofapplication

Role:
application
Description:

Deprecated. This symbol was intended to model MathML domainofapplication but as defined it is a synonym for domain. In MathML3, MathML compatibility is defined to use the new restriction symbol.

Signatures:
sts

 [Next: domain] [Previous: restriction] [Top]

## domain

Role:
application
Description:

This symbol denotes the domain of a given function, which is the set of values it is defined over.

Commented Mathematical property (CMP):
x is in the domain of f if and only if there exists a y in the range of f and f(x) = y
Formal Mathematical property (FMP):
$x\in \mathrm{domain}\left(f\right)\equiv \exists y.y\in \mathrm{range}\left(f\right)\wedge f\left(x\right)=y$
Signatures:
sts

 [Next: range] [Previous: domainofapplication] [Top]

## range

Role:
application
Description:

This symbol denotes the range of a function, that is a set that the function will map to. The single argument should be the function whos range is being queried. It should be noted that this is not necessarily equal to the image, it is merely required to contain the image.

Commented Mathematical property (CMP):
the range of f is a subset of the image of f
Formal Mathematical property (FMP):
$\mathrm{range}\left(f\right)\subset \mathrm{image}\left(f\right)$
Signatures:
sts

 [Next: image] [Previous: domain] [Top]

## image

Role:
application
Description:

This symbol denotes the image of a given function, which is the set of values the domain of the given function maps to.

Commented Mathematical property (CMP):
x in the domain of f implies f(x) is in the image f
Formal Mathematical property (FMP):
$x\in \mathrm{domain}\left(f\right)⇒f\left(x\right)\in \mathrm{image}\left(f\right)$
Signatures:
sts

 [Next: identity] [Previous: range] [Top]

## identity

Role:
application
Description:

The identity function, it takes one argument and returns the same value.

Commented Mathematical property (CMP):
for all x | identity(x)=x
Formal Mathematical property (FMP):
$\forall x.\mathrm{Id}\left(x\right)=x$
Signatures:
sts

 [Next: left_inverse] [Previous: image] [Top]

## left_inverse

Role:
application
Description:

This symbol is used to describe the left inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the left inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this left inverse.

Signatures:
sts

 [Next: right_inverse] [Previous: identity] [Top]

## right_inverse

Role:
application
Description:

This symbol is used to describe the right inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the right inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this right inverse.

Signatures:
sts

 [Next: inverse] [Previous: left_inverse] [Top]

## inverse

Role:
application
Description:

This symbol is used to describe the inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the inverse function is ill-defined without further stipulations. No assumptions are made on the semantics of this inverse.

Commented Mathematical property (CMP):
(inverse(f))(f(x)) = x if f is injective, that is (for all x,y | f(x) = f(y) implies x=y) implies (inverse(f))(f(z)) = z
Formal Mathematical property (FMP):
$\forall x,y.f\left(x\right)=f\left(y\right)⇒x=y⇒\left({f}^{-1}\right)\left(f\left(z\right)\right)=z$
Commented Mathematical property (CMP):
f(inverse(f(y))=y if f is defined at inverse(f)(y) that is, if there exists an x s.t. inverse(f)(y) = x then this implies f(inverse(f)(y)) = y
Formal Mathematical property (FMP):
$\exists x.\left({f}^{-1}\right)\left(y\right)=x⇒f\left(\left({f}^{-1}\right)\left(y\right)\right)=y$
Signatures:
sts

 [Next: left_compose] [Previous: right_inverse] [Top]

## left_compose

Role:
application
Description:

This symbol represents the function which forms the left-composition of its two (function) arguments.

Commented Mathematical property (CMP):
for all f,g,x | left_compose(f,g)(x) = f(g(x))
Formal Mathematical property (FMP):
$\forall f,g,x.\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)$
Signatures:
sts

 [Next: lambda] [Previous: inverse] [Top]

## lambda

Role:
binder
Description:

This symbol is used to represent anonymous functions as lambda expansions. It is used in a binder that takes two further arguments, the first of which is a list of variables, and the second of which is an expression, and it forms the function which is the lambda extraction of the expression

Example:
An example to show the connection between curried and uncurried applications of a binary function f (lambda(x,y).(f))(a,b)= (lambda(x).((lambda(y).(f))(b)))(a)
$\forall a,b.\lambda x,y.f=\lambda x.\lambda y.f$
Signatures:
sts

 [First: restriction] [Previous: left_compose] [Top]

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