Term syntax

Just as in mathematics an incredible number of fields — calculus, complex analysis, and so on — rest on a handful of axioms taken on faith, in the λ-calculus just three constructs form the foundation. The entire language is built from the variable, the abstraction, and the application; everything else — numbers, logic, data structures, and recursion — appears on top of them.

where $x$ is a variable; $(\lambda x.\,M)$ is an abstraction: an anonymous function with parameter $x$ and body $M$; $(M\,N)$ is an application: the function $M$ applied to the argument $N$. Here $N$ may be any term: a variable $x$, another application $(P\,Q)$, or a whole abstraction $(\lambda y.\,y)$. For example: $M\,x$, $M\,(P\,Q)$, $M\,(\lambda y.\,y)$.

A few important remarks: application is left-associative: $M\,N\,P$​${} \equiv ((M\,N)\,P)$; the body of an abstraction extends as far to the right as possible: $\lambda x.\,M\,N$​${} \equiv \lambda x.\,(M\,N)$; a chain of abstractions is abbreviated: $\lambda x\,y\,z.\,M$​${} \equiv \lambda x.\,\lambda y.\,\lambda z.\,M$.

A term is a tree: the leaves are variables, the inner nodes are abstractions and applications. It is worth training yourself to see a term as a tree rather than as a string.