# Definition: Function, Arity and Constant

let $L$ be a logical calculus, $U$ the domain of discourse, and $I(U,L)$ the corresponding interpretation with the valuation function $[[]]_I$.

A function is a non-empty string over an alphabet $s\in L$ interpreted as a function $$U^n\to U$$, taking $$n$$ input arguments from the domain of discourse and mapping them to a new value from the domain of discourse.

The natural number $$n\ge 0$$ of arguments of the function is called its arity. Special cases of arities are:

• $$n=0$$: Nullary functions (or operations) are usually called constants
• $$n=1$$: Unary functions (or operations)
• $$n=2$$: Binary functions (or operations)
• $$n=3$$: Ternary functions (or operations)
• generally $$n$$-nary functions (or operations)

Examples: 1

Definitions: 1 2 3
Examples: 4

Github: ### References

#### Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011