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 .
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
- In the (closed) formula , the variable is bound and not free.
- 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.
- In the (open) formula is bound and is free.
- In the formula for the class , the variable is bound and not free.
- In the formula for the power set , the variable is bound and free.
- When labeling , read as: “the one for which applies” (provided that it is unambiguous).
- 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 .
- 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):
|( Sum of finitely many values)
|is bound, and are free
|( Definite integral )
|is bound , and are free
|( Limit of an infinite sequence )
|is bound, is free
|( Limit value of a function at the point )
|is bound, and are free
- 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 .