# Operator

An operation $$f$$ on a set $$S$$ is a function that takes some values from $$S$$ and produces a new value. An operation can take any number of values from $$S$$, including zero (in which case $$f$$ is simply a constant) or infinitely many (in which case we call $$f$$ an “infinitary operation”). Common operations take a finite non-zero number of parameters. Operations often produce a value that is also in $$S$$ (in which case we say $$S$$ is closed under $$f$$), but that is not always the case.

For example, the function $$+$$ is a binary operation on $$\mathbb N$$, meaning it takes two values from $$\mathbb N$$ and produces another. Because $$+$$ produces a value that is also in $$\mathbb N$$, we say that $$\mathbb N$$ is closed under $$+$$.

The function $$\operatorname{neg}$$ that maps $$x$$ to $$-x$$ is a unary operation on $$\mathbb Z$$: It takes one value from $$\mathbb Z$$ as input, and produces an output in $$\mathbb Z$$ (namely, the negation of the input). $$\operatorname{neg}$$ is also a unary operation on $$\mathbb N$$, but $$\mathbb N$$ is not closed under $$\operatorname{neg}$$ (because $$\operatorname{neg}(3)=-3$$ is not in $$\mathbb N$$).

The number of values that the operator takes as input is called the arity of the operator. For example, the function $$\operatorname{zero}$$ which takes no inputs and returns $$0$$ is a zero-arity operator; and the operator $$f(a, b, c, d) = ac - bd$$ is a four-arity operator (which can be used on any ring, if we interpret multiplication and subtraction as ring operations).

Parents:

• What’s the difference between an operator and an operation?

Also what’s the difference between an operation and a function?