# Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I have also heard one of my professors mention in passing that an operation is a "special" kind of function.

Here's the thing, functions have only one output and therefore operations should too, however the indefinite integration operation has infinitely many outputs if the antiderivative exists. Can you please clear this up for me, perhaps the definition of an operation is wrong? And if so what is exactly an operation then? Thank you very much.

• Usually, an operation is a function : it takes as input one or more "objects", like numbers and produce an output, like addition or multiplication of two numbers. There are some "tricky cases" : division is normally understood as the "inverse" operation with respect to multiplication, but it is well known that $0/0$ is undefined, i.e. the "operation" $div$ is not defined for the input $(0,0)$. 1/2 Aug 29, 2015 at 15:11
• A similar case is the "sqaure root" operation; to raise a number to the power of two is a function, but its "inverse" is not, because roots of $4$ are $2$ and $-2$, and thus this "operation" is not a function. 2/2 Aug 29, 2015 at 15:11
• Totally depends on what you mean by "operation" and how you define a "function". It's more likely to be a "functional" - a function that returns a function. In that case, it's no problem to return a function with one extra parameter: the integration constant or something. Aug 29, 2015 at 15:11

In general, "operation" is an informal term, without a precise definition. Usually operations are functions, but as you point out with the example of antiderivatives, there are things that are sometimes referred to as operations which are not functions. If I had to give a generally applicable definition, I would say that an operation is a relation which is being thought of as if it were a function (i.e., something that takes an input and gives an output), even if it may not actually be a function (because it can actually give more than one output, or because it is not defined on some values that you might expect to be in its domain). In some contexts, though, an "operation" definitely does refer to a function. For instance, the term "binary operation" (or more generally, "$n$-ary operation") refers to a kind of function, as in Mauro Allegranza's answer.

Usually, like in algebra, an operation is a function.

See :

Definition 7.1.1. Let $A$ be a set. A binary operation on $A$ is a function $A \times A \to A$. A unary operation on $A$ is a function $A \to A$.