# Definition:Free Variable

## Definition

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.

## Examples

### Calculus Example

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

$x$ 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

- Definition:Free Occurrence: a somewhat more precise concept, recognising the fact that a variable may appear multiple times in an expression, and not necessarily always of the same category.

- Results about
**free variables**can be found here.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.4$: Universal and Existential Quantifiers - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 2$: The Axiom of Specification - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers - 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**free variable** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**variable**:**2.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**free variable** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**variable**:**2.**