Free variables and bound variables

In mathematics and logic is referred to as a variable in a mathematical formula freely occurring, if they do not occur in this formula at least one point in the range of an operator. However, all occurrences of the variable within the formula of operators are bonded, is referred to as the variable bound in this formula. A formula without free variables is closed formula, a formula having at least one free variable is called open formula.

For example, in predicate logic is a free individual variable in a predicate logic formula if they unquantified in this formula at least one location ( ie not in range of a quantifier to this variable) occurs. One with a quantifier (or) and used only within its binding domain variable is called bound. In predicate logic is a closed formula, ie a formula without free variables also called statement or sentence; an open formula, ie a formula with free variables is also called propositional form.

One and the same variable in a formula can have both free and bound occurrences. Knowledge of free and bound variables is required, for example for the cleanup of formulas.

Bound variables always come before in the notation of classes and quantities that are used in mathematics everywhere. They also occur in the lambda calculus and expressions with a bound variable of integration or summation variable.

Predicate logic definition

Examples

  • In the ( closed ) formula, the variable is bound and not free.
  • In the ( open ) formula, the variable occurs both bound and free before: Hardcover is their occurrence in the sub-formula, is free their occurrence in the subformula 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 of the variable is bound and free.

Other terms

  • Bound renaming: A bound by a quantifier variable can be replaced by another (not previously occurring ), with a logically equivalent formula is created. Example: From produced by bound renaming the formula.
  • Full Free Variable: A free variable without bound occurrences are also called fully free. By renaming bound can transform any formula into a logically equivalent, in which all free variables are actually fully free.

Mathematical notations with bound variables

In the following mathematical notations ( and many other ) a bound variable is used:

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