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Showing content from https://openmath.org/cd/fns1 below:

fns1

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.
       c) The derived work is distributed under terms that allow the
          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.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="fns1" name="domain"/>
      <OMA><OMS cd="fns1" name="restriction"/>
        <OMV name="f"/>
        <OMV name="S"/>
      </OMA>
    </OMA>
    <OMV name="S"/>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns1">domain</csymbol>
   <apply><csymbol cd="fns1">restriction</csymbol><ci>f</ci><ci>S</ci></apply>
  </apply>
  <ci>S</ci>
 </apply>
</math>

fns1.domain(fns1.restriction($f, $S)) = $S

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="fns1" name="restriction"/>
      <OMV name="f"/>
      <OMA><OMS cd="fns1" name="domain"/>
        <OMV name="f"/>
      </OMA>
    </OMA>
    <OMV name="f"/>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="fns1">restriction</csymbol>
   <ci>f</ci>
   <apply><csymbol cd="fns1">domain</csymbol><ci>f</ci></apply>
  </apply>
  <ci>f</ci>
 </apply>
</math>

fns1.restriction($f, fns1.domain($f)) = $f

Signatures:: sts

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

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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="equivalent"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS cd="fns1" name="domain"/>
	<OMV name="f"/>
      </OMA>
    </OMA>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="logic1" name="and"/>
        <OMA>
          <OMS cd="set1" name="in"/>
	  <OMV name="y"/>
	  <OMA>
            <OMS cd="fns1" name="range"/>
	    <OMV name="f"/>
	  </OMA>
        </OMA>
	<OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMV name="f"/>
	    <OMV name="x"/>
	  </OMA>
	  <OMV name="y"/>
	</OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">equivalent</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="fns1">domain</csymbol><ci>f</ci></apply>
  </apply>
  <bind><csymbol cd="quant1">exists</csymbol>
   <bvar><ci>y</ci></bvar>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="set1">in</csymbol>
     <ci>y</ci>
     <apply><csymbol cd="fns1">range</csymbol><ci>f</ci></apply>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><ci>f</ci><ci>x</ci></apply>
     <ci>y</ci>
    </apply>
   </apply>
  </bind>
 </apply>
</math>

logic1.equivalent(set1.in($x, fns1.domain($f)), quant1.exists[$y -> set1.in($y, fns1.range($f)) and $f($x) = $y])

x ∈ domain ⁡ ( f ) ≡ ∃ y . y ∈ range ⁡ ( f ) ∧ f ⁡ ( x ) = y

Signatures:: sts

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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="subset"/>
    <OMA>
      <OMS cd="fns1" name="range"/>
      <OMV name="f"/>
    </OMA>
    <OMA>
      <OMS cd="fns1" name="image"/>
      <OMV name="f"/>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">subset</csymbol>
  <apply><csymbol cd="fns1">range</csymbol><ci>f</ci></apply>
  <apply><csymbol cd="fns1">image</csymbol><ci>f</ci></apply>
 </apply>
</math>

set1.subset(fns1.range($f), fns1.image($f))

range ⁡ ( f ) ⊂ image ⁡ ( f )

Signatures:: sts

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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
	<OMS cd="fns1" name="domain"/>
	<OMV name="f"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMA>
        <OMV name="f"/>
	<OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="fns1" name="image"/>
	<OMV name="f"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>x</ci>
   <apply><csymbol cd="fns1">domain</csymbol><ci>f</ci></apply>
  </apply>
  <apply><csymbol cd="set1">in</csymbol>
   <apply><ci>f</ci><ci>x</ci></apply>
   <apply><csymbol cd="fns1">image</csymbol><ci>f</ci></apply>
  </apply>
 </apply>
</math>

set1.in($x, fns1.domain($f)) ==> set1.in($f($x), fns1.image($f))

x ∈ domain ⁡ ( f ) ⇒ f ⁡ ( x ) ∈ image ⁡ ( f )

Signatures:: sts

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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="x"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="fns1" name="identity"/>
        <OMV name="x"/>
      </OMA>
      <OMV name="x"/>
    </OMA>
  </OMBIND>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="fns1">identity</csymbol><ci>x</ci></apply>
   <ci>x</ci>
  </apply>
 </bind>
</math>

quant1.forall[$x -> fns1.identity($x) = $x]

Signatures:: sts

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

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

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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="x"/>
      <OMV name="y"/>
    </OMBVAR>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMV name="f"/>
	  <OMV name="x"/>
        </OMA>
        <OMA>
          <OMV name="f"/>
          <OMV name="y"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMA>
    </OMA>
  </OMBIND>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMA>
        <OMS cd="fns1" name="inverse"/>
        <OMV name="f"/>
      </OMA>
      <OMA>
        <OMV name="f"/>
        <OMV name="z"/>
      </OMA>
    </OMA>
    <OMV name="z"/>
  </OMA>
</OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <bind><csymbol cd="quant1">forall</csymbol>
   <bvar><ci>x</ci></bvar>
   <bvar><ci>y</ci></bvar>
   <apply><csymbol cd="logic1">implies</csymbol>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><ci>f</ci><ci>x</ci></apply>
     <apply><ci>f</ci><ci>y</ci></apply>
    </apply>
    <apply><csymbol cd="relation1">eq</csymbol><ci>x</ci><ci>y</ci></apply>
   </apply>
  </bind>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <apply><csymbol cd="fns1">inverse</csymbol><ci>f</ci></apply>
    <apply><ci>f</ci><ci>z</ci></apply>
   </apply>
   <ci>z</ci>
  </apply>
 </apply>
</math>

quant1.forall[$x, $y -> $f($x) = $f($y) ==> $x = $y] ==> fns1.inverse($f)($f($z)) = $z

∀ 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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="implies"/>
       <OMBIND>
        <OMS cd="quant1" name="exists"/>
	<OMBVAR>
	  <OMV name="x"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMA>
	      <OMS cd="fns1" name="inverse"/>
	      <OMV name="f"/>
	    </OMA>
	    <OMV name="y"/>
	  </OMA>
	  <OMV name="x"/>
	</OMA>
      </OMBIND>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMV name="f"/>
	<OMA>
	  <OMA>
	    <OMS cd="fns1" name="inverse"/>
	    <OMV name="f"/>
	  </OMA>
	  <OMV name="y"/>
	</OMA>
      </OMA>
      <OMV name="y"/>
    </OMA>
  </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <bind><csymbol cd="quant1">exists</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><apply><csymbol cd="fns1">inverse</csymbol><ci>f</ci></apply><ci>y</ci></apply>
    <ci>x</ci>
   </apply>
  </bind>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <ci>f</ci>
    <apply><apply><csymbol cd="fns1">inverse</csymbol><ci>f</ci></apply><ci>y</ci></apply>
   </apply>
   <ci>y</ci>
  </apply>
 </apply>
</math>

quant1.exists[$x -> fns1.inverse($f)($y) = $x] ==> $f(fns1.inverse($f)($y)) = $y

∃ x . ( f -1 ) ⁡ ( y ) = x ⇒ f ⁡ ( ( f -1 ) ⁡ ( y ) ) = y

Signatures:: sts

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):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="f"/>
    <OMV name="g"/>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMA>
        <OMS cd="fns1" name="left_compose"/>
        <OMV name="f"/>
        <OMV name="g"/>
      </OMA>
      <OMV name="x"/>
    </OMA>
    <OMA>
      <OMV name="f"/>
      <OMA>
        <OMV name="g"/>
        <OMV name="x"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>f</ci></bvar>
  <bvar><ci>g</ci></bvar>
  <bvar><ci>x</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <apply><csymbol cd="fns1">left_compose</csymbol><ci>f</ci><ci>g</ci></apply>
    <ci>x</ci>
   </apply>
   <apply><ci>f</ci><apply><ci>g</ci><ci>x</ci></apply></apply>
  </apply>
 </bind>
</math>

quant1.forall[$f, $g, $x -> fns1.left_compose($f, $g)($x) = $f($g($x))]

∀ f , g , x . ( f o g ) ⁡ ( x ) = f ⁡ ( g ⁡ ( x ) )

Signatures:: sts

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)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="a"/>
    <OMV name="b"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMBVAR>
        <OMV name="f"/>
      </OMBIND>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMA>
    <OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="x"/>
        </OMBVAR>
        <OMA>
          <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="y"/>
            </OMBVAR>
            <OMV name="f"/>
          </OMBIND>
          <OMV name="b"/>
        </OMA>
      </OMBIND>
      <OMV name="a"/>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <bvar><ci>y</ci></bvar>
     <ci>f</ci>
    </bind>
    <ci>a</ci>
    <ci>b</ci>
   </apply>
   <apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <apply>
      <bind><csymbol cd="fns1">lambda</csymbol><bvar><ci>y</ci></bvar><ci>f</ci></bind>
      <ci>b</ci>
     </apply>
    </bind>
    <ci>a</ci>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$a, $b -> fns1.lambda[$x, $y -> $f]($a, $b) = fns1.lambda[$x -> fns1.lambda[$y -> $f]($b)]($a)]

∀ a , b . λ x , y . f = λ x . λ y . f

Signatures:: sts

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