Namespaces
Variants
Actions

Difference between revisions of "Free variable"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''free occurrence of a variable''
+
<!--
 +
f0416601.png
 +
$#A+1 = 36 n = 0
 +
$#C+1 = 36 : ~/encyclopedia/old_files/data/F041/F.0401660 Free variable,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The occurrence of a variable in a linguistic expression as a parameter of this expression. A rigorous definition of this concept can be given only for a [[Formalized language|formalized language]]. Every language has its own definition of a free variable, depending on the rules for forming expressions in the particular language. The semantic criterion is the following condition: The substitution of any object from an implicit interpretation in place of the given occurrence(s) of a variable must not lead to an absurd expression. For example, in the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416601.png" />, which denotes the set of points of a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416602.png" />, the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416603.png" /> is free while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416605.png" /> are not (see [[Bound variable|Bound variable]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416606.png" /> denotes a mapping of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416607.png" />, and the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f0416609.png" /> range over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166011.png" />, respectively, then in the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166012.png" /> the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166014.png" /> are free (and so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166015.png" />, if it is considered as a variable with respect to functions). For a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166016.png" /> and by varying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166017.png" /> one obtains a function of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166018.png" />, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166019.png" />. In this expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166020.png" /> is free and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166021.png" /> is not. In the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166022.png" />, which denotes the value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166023.png" /> at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166024.png" />, the last occurrence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166025.png" /> is free while the two others are not. The first occurrence is called an operator occurrence (it is under the sign of an operator), and the second a bound occurrence.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
For a non-formalized language, that is, in actual mathematical texts, for an individual expression it is not always possible to definitely identify the free variables and the bound ones. For example, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166026.png" />, depending on the context, the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166027.png" /> can be free and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166028.png" /> bound, or vice-versa, but they cannot both be free. An indication of which variable is assumed to be free is given by using additional means. For example, if this expression is met in a context of the form  "let fk=i&lt; kaik" , then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166029.png" /> is free. If there is agreement that there is no summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166031.png" /> is a parameter. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166032.png" />, often used in mathematics, sometimes denotes a one-element set, in which case the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166033.png" /> occurs freely, and sometimes denotes the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166034.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166035.png" /> runs over an assigned domain of objects, in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041660/f04166036.png" /> is a bound variable.
+
''free occurrence of a variable''
  
 +
The occurrence of a variable in a linguistic expression as a parameter of this expression. A rigorous definition of this concept can be given only for a [[Formalized language|formalized language]]. Every language has its own definition of a free variable, depending on the rules for forming expressions in the particular language. The semantic criterion is the following condition: The substitution of any object from an implicit interpretation in place of the given occurrence(s) of a variable must not lead to an absurd expression. For example, in the expression  $  \{ {( x, y) } : {x  ^ {2} + y  ^ {2} = z  ^ {2} } \} $,
 +
which denotes the set of points of a circle of radius  $  z $,
 +
the variable  $  z $
 +
is free while  $  x $
 +
and  $  y $
 +
are not (see [[Bound variable|Bound variable]]). If  $  f $
 +
denotes a mapping of the form  $  X \times Y \rightarrow Z $,
 +
and the variables  $  x $
 +
and  $  y $
 +
range over  $  X $
 +
and  $  Y $,
 +
respectively, then in the expression  $  f ( x, y) $
 +
the variables  $  x $
 +
and  $  y $
 +
are free (and so is  $  f $,
 +
if it is considered as a variable with respect to functions). For a fixed  $  x $
 +
and by varying  $  y $
 +
one obtains a function of the form  $  Y \rightarrow Z $,
 +
which is denoted by  $  \lambda yf ( x, y) $.
 +
In this expression  $  x $
 +
is free and  $  y $
 +
is not. In the expression  $  ( \lambda yf ( x, y)) ( y) $,
 +
which denotes the value of the function  $  \lambda yf ( x, y) $
 +
at an arbitrary point  $  y $,
 +
the last occurrence of  $  y $
 +
is free while the two others are not. The first occurrence is called an operator occurrence (it is under the sign of an operator), and the second a bound occurrence.
  
 +
For a non-formalized language, that is, in actual mathematical texts, for an individual expression it is not always possible to definitely identify the free variables and the bound ones. For example, in  $  \sum _ {i < k }  a _ {ik} $,
 +
depending on the context, the variable  $  i $
 +
can be free and  $  k $
 +
bound, or vice-versa, but they cannot both be free. An indication of which variable is assumed to be free is given by using additional means. For example, if this expression is met in a context of the form  "let fk=i&lt; kaik" , then  $  k $
 +
is free. If there is agreement that there is no summation over  $  k $,
 +
then  $  k $
 +
is a parameter. The expression  $  \{ a _ {i} \} $,
 +
often used in mathematics, sometimes denotes a one-element set, in which case the variable  $  i $
 +
occurs freely, and sometimes denotes the set of all  $  a _ {i} $
 +
where  $  i $
 +
runs over an assigned domain of objects, in which case  $  i $
 +
is a bound variable.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


free occurrence of a variable

The occurrence of a variable in a linguistic expression as a parameter of this expression. A rigorous definition of this concept can be given only for a formalized language. Every language has its own definition of a free variable, depending on the rules for forming expressions in the particular language. The semantic criterion is the following condition: The substitution of any object from an implicit interpretation in place of the given occurrence(s) of a variable must not lead to an absurd expression. For example, in the expression $ \{ {( x, y) } : {x ^ {2} + y ^ {2} = z ^ {2} } \} $, which denotes the set of points of a circle of radius $ z $, the variable $ z $ is free while $ x $ and $ y $ are not (see Bound variable). If $ f $ denotes a mapping of the form $ X \times Y \rightarrow Z $, and the variables $ x $ and $ y $ range over $ X $ and $ Y $, respectively, then in the expression $ f ( x, y) $ the variables $ x $ and $ y $ are free (and so is $ f $, if it is considered as a variable with respect to functions). For a fixed $ x $ and by varying $ y $ one obtains a function of the form $ Y \rightarrow Z $, which is denoted by $ \lambda yf ( x, y) $. In this expression $ x $ is free and $ y $ is not. In the expression $ ( \lambda yf ( x, y)) ( y) $, which denotes the value of the function $ \lambda yf ( x, y) $ at an arbitrary point $ y $, the last occurrence of $ y $ is free while the two others are not. The first occurrence is called an operator occurrence (it is under the sign of an operator), and the second a bound occurrence.

For a non-formalized language, that is, in actual mathematical texts, for an individual expression it is not always possible to definitely identify the free variables and the bound ones. For example, in $ \sum _ {i < k } a _ {ik} $, depending on the context, the variable $ i $ can be free and $ k $ bound, or vice-versa, but they cannot both be free. An indication of which variable is assumed to be free is given by using additional means. For example, if this expression is met in a context of the form "let fk=i< kaik" , then $ k $ is free. If there is agreement that there is no summation over $ k $, then $ k $ is a parameter. The expression $ \{ a _ {i} \} $, often used in mathematics, sometimes denotes a one-element set, in which case the variable $ i $ occurs freely, and sometimes denotes the set of all $ a _ {i} $ where $ i $ runs over an assigned domain of objects, in which case $ i $ is a bound variable.

Comments

References

[a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
How to Cite This Entry:
Free variable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_variable&oldid=46988
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article