The Relation, Function & Operation

[questar]

This thread does justice to a question put forth online several times and, as far as I can tell is only answered in part. I believe this question warrants a distinct and succinct answer. What I'm finding online is summarized below, and as one can see... there is something missing.

I've been thinking about the primary similarities and differences between the relation, function and operation.

For instance, a function is always a relation, but a relation is not necessarily a function.
A relation is not necessarily a function because a relation,
unlike a function, may involve more than one output.

In the same vein it can also be said that an operation is always a function, but a function is not necessarily an operation.
A function is not necessarily an operation because a function,
unlike an operation, ______________________________.

When is an operation not a function?

 

[jbunniii]

I don't think there is a meaningful distinction to be made here. An operation is a function, and a function is an operation. They are two words for the same thing.

It is true that we are more likely to use the word "operation" for some kinds of functions, such as the addition function $+ : \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$. Similarly we might be more likely to say "map" when dealing with linear functions. But all of these words are interchangeable.

 

[micromass]

I don't think there is a meaningful distinction to be made here. An operation is a function, and a function is an operation. They are two words for the same thing.

It is true that we are more likely to use the word "operation" for some kinds of functions, such as the addition function $+ : \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$. Similarly we might be more likely to say "map" when dealing with linear functions. But all of these words are interchangeable.

I have never heard of an "operation" being a synonym of a "function".

 

[disregardthat]

An operation on pairs of sets, such as the cartesian product, can be seen as a "function" in one sense.

For each pair of sets V and W, we may define the cartesian product $V \times W$. This is an operation in the conventional use of the word, but it is not actually a function at all. The reason is that you can't form the set of sets, and by insisting on a function to be a set itself, it is impossible to have the cartesian product operation as a function.

The only times the words function and operation are used interchangably is when they actually are synonymous. This is however not always the case.

 

[micromass]

An operation on pairs of sets, such as the cartesian product, can be seen as a "function" in one sense.

For each pair of sets V and W, we may define the cartesian product [itex]V \times W[\itex]. This is an operation in the conventional use of the word, but it is not actually a function at all. The reason is that you can't form the set of sets, and by insisting on a function to be a set itself, it is impossible to have the cartesian product operation as a function.

Set theory texts do allow proper classes to be domain and codomain of functions and typically do call them function/operator.

 

[disregardthat]

Set theory texts do allow proper classes to be domain and codomain of functions and typically do call them function/operator.

Usually in the language of category theory, the word "functor" is used. Or, if we are talking about objects, "morphisms".

 

[micromass]

Usually in category theory, the word "functor" is used.

A functor is something completely different as a function between proper classes. A functor specifies a map between the object classes and a map between the morphism classes.

 

[jbunniii]

I have never heard of an "operation" being a synonym of a "function".

Well, if we start with the ever-authoritative Wikipedia :tongue2: then we have

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values, called "operands"

which sounds to me like some function $f : \prod_{n=1}^{N} A_n \rightarrow B$. Conversely, if we have an arbitrary function $f : A \rightarrow B$ then it's not unheard of to refer to "the operation of applying $f$ to $a \in A$".

However, others have made good points, and another distinction also occurred to me: we may talk about the "operation of addition" without referring to the underlying set, so in some sense it can refer to a class of functions (addition of integers, rationals, reals, whatever). So I retract my previous assertion of synonmity.

 

[DavidHume]

The fuzzy thinking on this distinction is exacerbated by the sloppy terminology of (most) computer languages which distinguish between a function and an operator based solely on the symbols used. So, in computer languages, "+" is called an operator but "plus" would be a function even if "plus" were used exactly the same way as "+" and returned exactly the same results.

This is a very simple but not particularly useful distinction, to say that something is a function if its name is made up of alphanumeric symbols but is an operator if it uses non-alphanumeric symbols.

A more useful distinction, made in mathematics and preserved in maybe two computer languages (APL, J), is that a function takes data as its arguments, e.g. "+" in "1+2" or "plus" in "1 plus 2", whereas an operator takes both data and one or more functions as its arguments, as in the integration of a function between certain limits.

So, multiplication and subtraction are functions but differentiation and integration are operators. This introduces a potential usefully hierarchy and distinction between fundamentally distinct concepts, unlike the trivial and unhelpful distinction implied by most computer languages.