# Free variable

The occurrence of a variable in a linguistic expression as a parameter of this expression. A rigorous definition of this concept can be given only for a formalized language. Every language has its own definition of a free variable, depending on the rules for forming expressions in the particular language. The semantic criterion is the following condition: The substitution of any object from an implicit interpretation in place of the given occurrence(s) of a variable must not lead to an absurd expression. For example, in the expression $\{ {( x, y) } : {x ^ {2} + y ^ {2} = z ^ {2} } \}$, which denotes the set of points of a circle of radius $z$, the variable $z$ is free while $x$ and $y$ are not (see Bound variable). If $f$ denotes a mapping of the form $X \times Y \rightarrow Z$, and the variables $x$ and $y$ range over $X$ and $Y$, respectively, then in the expression $f ( x, y)$ the variables $x$ and $y$ are free (and so is $f$, if it is considered as a variable with respect to functions). For a fixed $x$ and by varying $y$ one obtains a function of the form $Y \rightarrow Z$, which is denoted by $\lambda yf ( x, y)$. In this expression $x$ is free and $y$ is not. In the expression $( \lambda yf ( x, y)) ( y)$, which denotes the value of the function $\lambda yf ( x, y)$ at an arbitrary point $y$, the last occurrence of $y$ is free while the two others are not. The first occurrence is called an operator occurrence (it is under the sign of an operator), and the second a bound occurrence.
For a non-formalized language, that is, in actual mathematical texts, for an individual expression it is not always possible to definitely identify the free variables and the bound ones. For example, in $\sum _ {i < k } a _ {ik}$, depending on the context, the variable $i$ can be free and $k$ bound, or vice-versa, but they cannot both be free. An indication of which variable is assumed to be free is given by using additional means. For example, if this expression is met in a context of the form "let fk=i< kaik" , then $k$ is free. If there is agreement that there is no summation over $k$, then $k$ is a parameter. The expression $\{ a _ {i} \}$, often used in mathematics, sometimes denotes a one-element set, in which case the variable $i$ occurs freely, and sometimes denotes the set of all $a _ {i}$ where $i$ runs over an assigned domain of objects, in which case $i$ is a bound variable.