# 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 .

## 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}$

## Web links

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