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


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  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
            

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):
domain ( f | S ) = S
Formal Mathematical property (FMP):
f | domain ( f ) = 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 domain ( f ) y . y range ( f ) f ( x ) = 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):
range ( f ) image ( f )
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 domain ( f ) f ( x ) image ( f )
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):
x . Id ( x ) = 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):
x , y . f ( x ) = f ( y ) x = y ( f -1 ) ( f ( z ) ) = 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):
x . ( f -1 ) ( y ) = x f ( ( f -1 ) ( y ) ) = 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):
f , g , x . ( f o g ) ( x ) = f ( g ( x ) )
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)
a , b . λ x , y . f = λ x . λ y . f
Signatures:
sts


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