Free variable and bound variable

In mathematics and logic , a variable is referred to as occurring freely in a mathematical formula if it occurs in this formula in at least one place that is not in the area of an operator . If, on the other hand, all occurrences of the variable within the formula are linked to operators, the variable is said to be linked in this formula . A formula without free variables is called a closed formula , a formula with at least one free variable is called an open formula .

For example, in predicate logic, an individual variable in a predicate logic formula is free if it occurs in this formula in at least one place unquantified (i.e. not in the area of a quantifier for this variable). A variable used with a quantifier ( or ) and only within its binding area is called bound . In predicate logic, a closed formula, that is, a formula without free variables, is also called a statement or sentence ; an open formula, i.e. a formula with free variables, is also called a statement form . ${\ displaystyle \ forall}$ ${\ displaystyle \ exists}$

The same variable can have both free and bound occurrences in a formula. Knowledge of free and bound variables is required , for example, to clean up formulas.

Bound variables always appear in the notation of classes and sets , which are used everywhere in mathematics. They also occur in the lambda calculus and in expressions with a bound integration variable or summation variable as well as in identifications .

Predicate logic definition

Examples

In the (closed) formula , the variable is bound and not free. ${\ displaystyle \ forall xP (x)}$ ${\ displaystyle x}$
In the (open) formula , the variable occurs both bound and free: its occurrence is bound in the partial formula , its occurrence in the partial formula is free , to which the universal quantifier no longer extends. ${\ displaystyle \ forall xP (x) \ land Q (x)}$ ${\ displaystyle x}$ ${\ displaystyle \ forall xP (x)}$ ${\ displaystyle Q (x)}$
In the (open) formula is bound and is free. ${\ displaystyle \ forall x (P (x) \ land Q (x, y))}$ ${\ displaystyle x}$ ${\ displaystyle y}$
In the formula for the class , the variable is bound and not free. ${\ displaystyle \ {x \ mid A (x) \}}$ ${\ displaystyle x}$
In the formula for the power set , the variable is bound and free. ${\ displaystyle {\ mathcal {P}} a = \ {x \ mid x \ subseteq a \}}$ ${\ displaystyle x}$ ${\ displaystyle \, a}$
When labeling , read as: “the one for which applies” (provided that it is unambiguous). ${\ displaystyle \ iota xF (x)}$ ${\ displaystyle x}$ ${\ displaystyle F (x)}$

More terms

Bound renaming : A variable bound by a quantifier can be replaced by another (which did not appear before), resulting in a logically equivalent formula. Example: The formula arises from a bound renaming .

{\ displaystyle \ forall xP (x) \ land Q (x, y)}

{\ displaystyle \ forall zP (z) \ land Q (x, y)}

Completely free variable : A free variable with no bound occurrence is also called fully free. Any formula can be converted into a logically equivalent one by means of bound renaming, in which all free variables are actually completely free.

Mathematical notations with bound variables

A bound variable is used in the following math notations (and many more):

${\ displaystyle \ sum _ {i = 1} ^ {n} a_ {i}}$	( Sum of finitely many values)	${\ displaystyle i}$ is bound, and are free ${\ displaystyle n}$ ${\ displaystyle a}$
${\ displaystyle \ int _ {a} ^ {b} f (x) \, \ mathrm {d} x}$	( Definite integral )	${\ displaystyle x}$ is bound , and are free ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle f}$
${\ displaystyle \ lim _ {n \ to \ infty} a_ {n}}$	( Limit of an infinite sequence )	${\ displaystyle n}$ is bound, is free ${\ displaystyle a}$
${\ displaystyle \ lim _ {x \ to x_ {0}} f (x)}$	( Limit value of a function at the point ) ${\ displaystyle x_ {0}}$	${\ displaystyle x}$ is bound, and are free ${\ displaystyle x_ {0}}$ ${\ displaystyle f}$

literature

Wolfgang Rautenberg : Introduction to Mathematical Logic . 3. Edition. Vieweg + Teubner, Wiesbaden 2008, ISBN 978-3-8348-0578-2 .
H.-P. Tuschik, H. Wolter: Mathematical Logic - In Brief . Spectrum, Akad. Verlag, Heidelberg 2002, ISBN 3-8274-1387-7 .

Web links

Wikibooks: Math for Non-Freaks: Free and Tied Variables - Learning and Teaching Materials