Lambda Variable

lambda-calculus

Definition

Lambda Variable

The set of variable symbols is defined as :

$$\mathcal{V}=\{a,b,c,\dots ,x,y,z,x^{\prime },y^{\prime },z^{\prime },\dots ,x^{\prime \prime },\dots \}$$

Bound

Bound Lambda Variable

A variable is called bound if it is introduced by a $\lambda$ within the expression and falls within the scope of that $\lambda$.

Examples:

• $\lambda \underline{x}.\underline{x}$: both instances of $x$ are bound by the $\lambda x$.

Free

Free Lambda Variable

A variable is called free if it appears in the expression but it is not bound by any enclosing $\lambda$ within that specific expression.

Examples:

• $\lambda x.\underline{y}$: the variable $y$ is free because there’s no $\lambda y$ binding it.