Namespaces
Variants
Actions

Difference between revisions of "Bound variable"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX done)
 
Line 2: Line 2:
  
 
A type of occurrence of a variable in a linguistic expression. An exact definition for each formalized language depends on the rules of formation of the language. It is not possible to substitute objects in place of a bound variable, since such a substitution would lead to meaningless expressions. But the replacement of a bound variable, where it occurs, by a new (for a given expression) variable leads to an expression with the same meaning. For example, in the expressions
 
A type of occurrence of a variable in a linguistic expression. An exact definition for each formalized language depends on the rules of formation of the language. It is not possible to substitute objects in place of a bound variable, since such a substitution would lead to meaningless expressions. But the replacement of a bound variable, where it occurs, by a new (for a given expression) variable leads to an expression with the same meaning. For example, in the expressions
 +
$$
 +
\int f(x,y)\,d x\,,\ \ \ \{ x : f(x,y) = 0 \}\,,
 +
$$
 +
$x$ is a bound variable. Replacing $x$ by a number leads to a meaningless expression, whereas by writing $z$ everywhere instead of $z$ one obtains an expression with exactly the same meaning.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172301.png" /></td> </tr></table>
+
Bound variables always arise in applying to an expression $\mathcal{E}$ with free occurrences of a variable $x$ an operator variable $x$ (see [[Free variable]]). In the resulting expression, all the occurrences of $x$ in $\mathcal{E}$ that were previously free become bound. We mention below certain operators that are often used (next to the operators $\int \ldots dx$ and $\{ x : \ldots \}$), in which $x$ is an operator variable:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172302.png" /> is a bound variable. Replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172303.png" /> by a number leads to a meaningless expression, whereas by writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172304.png" /> everywhere instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172305.png" /> one obtains an expression with exactly the same meaning.
+
$\forall x (\ldots)$, $\exists x (\ldots)$, that is, the universal and existential quantifiers;
  
Bound variables always arise in applying to an expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172306.png" /> with free occurrences of a variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172307.png" /> an operator variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172308.png" /> (see [[Free variable|Free variable]]). In the resulting expression, all the occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b0172309.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723010.png" /> that were previously free become bound. We mention below certain operators that are often used (next to the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723012.png" />), in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723013.png" /> is an operator variable:
+
$\int_{\ldots}^{\ldots} \ldots dx$, that is, a definite integral with respect to $x$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723014.png" />, that is, the universal and existential quantifiers;
+
$\sum_x$ that is, summation over $x$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723015.png" />, that is, a definite integral with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723016.png" />;
+
$\lambda x . (\ldots)$, that is, a function of $x$ the value of which at $x$ is $\ldots$. Specific linguistic expressions can be substituted in place of the dots.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723017.png" /> that is, summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723018.png" />;
+
In real (non-formalized) mathematical texts it is possible to have a non-unique use for one and the same expression; in this connection distinguishing a bound variable in a given expression depends on the context and meaning of the expression. In formalized languages there is a formal procedure for distinguishing free and bound occurrences of variables.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723019.png" />, that is, a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723020.png" /> the value of which at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723021.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017230/b01723022.png" />. Specific linguistic expressions can be substituted in place of the dots.
+
{{TEX|done}}
 
 
In real (non-formalized) mathematical texts it is possible to have a non-unique use for one and the same expression; in this connection distinguishing a bound variable in a given expression depends on the context and meaning of the expression. In formalized languages there is a formal procedure for distinguishing free and bound occurrences of variables.
 

Latest revision as of 21:45, 2 March 2018

bound occurrence of a variable

A type of occurrence of a variable in a linguistic expression. An exact definition for each formalized language depends on the rules of formation of the language. It is not possible to substitute objects in place of a bound variable, since such a substitution would lead to meaningless expressions. But the replacement of a bound variable, where it occurs, by a new (for a given expression) variable leads to an expression with the same meaning. For example, in the expressions $$ \int f(x,y)\,d x\,,\ \ \ \{ x : f(x,y) = 0 \}\,, $$ $x$ is a bound variable. Replacing $x$ by a number leads to a meaningless expression, whereas by writing $z$ everywhere instead of $z$ one obtains an expression with exactly the same meaning.

Bound variables always arise in applying to an expression $\mathcal{E}$ with free occurrences of a variable $x$ an operator variable $x$ (see Free variable). In the resulting expression, all the occurrences of $x$ in $\mathcal{E}$ that were previously free become bound. We mention below certain operators that are often used (next to the operators $\int \ldots dx$ and $\{ x : \ldots \}$), in which $x$ is an operator variable:

$\forall x (\ldots)$, $\exists x (\ldots)$, that is, the universal and existential quantifiers;

$\int_{\ldots}^{\ldots} \ldots dx$, that is, a definite integral with respect to $x$;

$\sum_x$ that is, summation over $x$;

$\lambda x . (\ldots)$, that is, a function of $x$ the value of which at $x$ is $\ldots$. Specific linguistic expressions can be substituted in place of the dots.

In real (non-formalized) mathematical texts it is possible to have a non-unique use for one and the same expression; in this connection distinguishing a bound variable in a given expression depends on the context and meaning of the expression. In formalized languages there is a formal procedure for distinguishing free and bound occurrences of variables.

How to Cite This Entry:
Bound variable. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bound_variable&oldid=14900
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article