In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A formal language is a set of words, ie finite strings of letters, or symbols. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In mathematics the word expression is a term for any well-formed combination of mathematical symbols First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol. The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece A string literal is the representation of a string value within the Source code of a Computer program. For other meanings of 'wild card' see Wild card. The term wildcard character has the following meanings Telecommunication In

The variable x becomes a bound variable, for example, when we write

'For all x, (x + 1)2 = x2 + 2x + 1. '

or

'There exists x such that x2 = 2. '

In either of these propositions, it does not matter logically whether we use x or some other letter. However, it could be confusing to use the same letter again elsewhere in some compound proposition. In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence That is, free variables become bound, and then in a sense retire from further work supporting the formation of formulae.

In computer programming, a free variable is a variable referred to in a function that is not a local variable or an argument of that function. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. In Computer science, a local variable is a Variable that is given local scope. In Computer programming, a parameter is a variable which takes on the meaning of a corresponding Argument (computer science is same article--> argument In Computer science, a subroutine ( function, method, procedure, or subprogram) is a portion of code within a larger

## Examples

Before stating a precise definition of free variable and bound variable (or dummy variable), we present some examples that perhaps make these two concepts clearer than the definition would (unfortunately the term dummy variable is used by many statisticians to mean an indicator variable or some variant thereof; the name is really not apt for that purpose, but magnificently conveys the intuition behind the definition of this concept):

In the expression

$\sum_{k=1}^{10} f(k,n),$

n is a free variable and k is a bound variable (or dummy variable); consequently the value of this expression depends on the value of n, but there is nothing called k on which it could depend. In Regression analysis, a dummy variable (also known as indicator variable or just dummy) is one that takes the values 0 or 1 to indicate the absence

In the expression

$\int_0^\infty x^{y-1} e^{-x}\,dx,$

y is a free variable and x is a bound variable; consequently the value of this expression depends on the value of y, but there is nothing called x on which it could depend.

In the expression

$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$

x is a free variable and h is a bound variable; consequently the value of this expression depends on the value of x, but there is nothing called h on which it could depend.

In the expression

$\forall x\ \exists y\ \varphi(x,y,z),$

z is a free variable and x and y are bound variables; consequently the logical value of this expression depends on the value of z, but there is nothing called x or y on which it could depend. In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true

### Variable-binding operators

The following

 $\sum_{x\in S}$ $\int_0^\infty\cdots\,dx$ $\lim_{x\to 0}$ $\forall x$ $\exists x$

are variable-binding operators. Each of them binds the variable x.

## Formal explanation

Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science but in all cases they are purely syntactic properties of expressions and variables in them. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the For this section we can summarize syntax by identifying an expression with a tree whose leaf nodes are variables, constants, function constants or predicate constants and whose non-leaf nodes are logical operators. A tree is a perennial Woody plant. It is most often defined as a woody plant that has many secondary branches supported clear of the ground on a single main stem or Variable-binding operators are logical operators that occur in almost every formal language. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence Indeed languages which do not have them are either extremely inexpressive or extremely difficult to use. A binding operator Q takes two arguments: a variable v and an expression P, and when applied to its arguments produces a new expression Q(v, P). The meaning of binding operators is supplied by the semantics of the language and does not concern us here. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from

$\forall x\, (\exists y\, A(x) \vee B(z))$

Variable binding relates three things: a variable v, a location a for that variable in an expression and a non-leaf node n of the form Q(v, P). Note: we define a location in an expression as a leaf node in the syntax tree. Variable binding occurs when that location is below the node n

To give an example from mathematics, consider an expression which defines a function

$(x_1, \ldots , x_n) \mapsto \operatorname{t}$

where t is an expression. t may contain some, all or none of the x1, . . . , xn and it may contain other variables. In this case we say that function definition binds the variables x1, . . . , xn.

In the lambda calculus, x is a bound variable in the term M = λ x . In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function T, and a free variable of T. We say x is bound in M and free in T. If T contains a subterm λ x . U then x is rebound in this term. This nested, inner binding of x is said to "shadow" the outer binding. Occurrences of x in U are free occurrences of the new x.

Variables bound at the top level of a program are technically free variables within the terms to which they are bound but are often treated specially because they can be compiled as fixed addresses. Similarly, an identifier bound to a recursive function is also technically a free variable within its own body but is treated specially. Computable functions are the basic objects of study in computability theory.

A closed term is one containing no free variables.