Definition:Bound Variable
Definition
A bound variable is a variable which, when it occurs in an expression, can be replaced with another variable without changing the meaning of the statement.
Examples
Algebraic Example
In algebra:
- $x^2 + 2 x y + y^2 = \paren {x + y}^2$
both $x$ and $y$ are bound variables.
Universal Statement
In the universal statement:
- $\forall x: \map P x$
the symbol $x$ is a bound variable.
Thus, the meaning of $\forall x: \map P x$ does not change if $x$ is replaced by another symbol.
That is, $\forall x: \map P x$ means the same thing as $\forall y: \map P y$ or $\forall \alpha: \map P \alpha$.
And so on.
Existential Statement
In the existential statement:
- $\exists x: \map P x$
the symbol $x$ is a bound variable.
Thus, the meaning of $\exists x: \map P x$ does not change if $x$ is replaced by another symbol.
That is, $\exists x: \map P x$ means the same thing as $\exists y: \map P y$ or $\exists \alpha: \map P \alpha$. And so on.
Family of Sets
Let $I$ be an indexing set.
Consider the union of the indexed family of sets $\family {S_i}_{i \mathop \in I}$:
- $\ds \bigcup_{i \mathop \in I} S_i$
The variable $i$ is a bound variable, or dummy variable, such that $\ds \bigcup_{i \mathop \in I} S_i$ could as well be written $\ds \bigcup_{\alpha \mathop \in I} S_\alpha$ or $\ds \bigcup_{\gamma \mathop \in I} S_\gamma$, for example.
Also known as
A bound variable is also popularly seen with the name dummy variable. Both terms can be seen on $\mathsf{Pr} \infty \mathsf{fWiki}$.
In treatments of pure logic, this is sometimes known as an individual variable.
Some sources call it an apparent variable, reflecting the fact that it only "appears" to be a variable.
Also see
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
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Sets
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 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
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 3$: Quantifiers
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability: $\S 2.1$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: bound: 4.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: bound: 4.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: dummy variable