Definition:Free Variable

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Let $x$ be a variable in an expression $E$.

$x$ is a free variable in $E$ if and only if it is not a bound variable.

Predicate Logic

In the context of predicate logic, the concept has a precise definition:

In predicate logic, a free variable is a variable which exists in a WFF only as free occurrences.


Calculus Example

In differential calculus:

$\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

$x$ is a free variable, as a function's derivative varies with the input being considered.

Cardinality Example

In set theory:

$\card S = \aleph_0$

$S$ is a free variable, as, for instance, $S = \Z$ makes this true while $S = \R$ makes it false.

Series Example

In the inequality:

$\ds \sum_{n \mathop = 0}^\infty a z^n < z^2$

$a$ and $z$ are both free variables, as the inequality may or may not hold depending on their values.

Also known as

A free variable is often referred to as an unknown, particularly in mathematical contexts.

In the field of logic, a free variable can also be referred to as a real variable.

However, this can be confused with a variable whose domain is the set of real numbers, so its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is discouraged.

The name arises in apposition to the name apparent variable, which is another name for bound variable.

Also see

  • Results about free variables can be found here.