Definitions of operation and function

[Rasalhague]

Definitions of "operation" and "function"

Is every operation a function? Is every function an operation? From the definitions I've read, I'm guessing yes. If not, what would be an example of a function that isn't an operation, or an operation that isn't a function?

 

[Rasalhague]

Is there a term that means specifically the "rule" part of a function, the condition which determines which elements of the domain and codomain are paired?

 

[Elucidus]

I have seen the word "operator" used to be synonymous with "function," as well as "mapping," "transformation," and "correspondence."

For example the "limit operator," "logical operator."

But I have also seen the word "operator" to refer to a specific kind of function:

An operator is a function f:A^n\rightarrow A where A is a set. The arity of f is defined to be n. In this case f acts on ordered n-tuples. The key restriction is that it must produce a value in the same set (forcing it to be closed).

Some examples:

+=((a,b)\mapsto (a+b)):\mathbb{R}^2\rightarrow\mathbb{R}

\langle\;\rangle=\left((x_1,x_2,x_3)\mapsto \sqrt{x_1^2+x_2^2+x_3^2}\right):\mathbb{R}^3\rightarrow\mathbb{R}

The part of the function that describes the correspondence is often called the "rule of assignment." It is sometimes written (x\mapsto y) for appropriate x and y.

 

[Rasalhague]

Thanks, Elucidus. Is the term "ordered n-tuple" synonymous with "n-tuple"? Your special definition of "operator" sounds a lot like the definitions of "operation" at Wikipedia and Wolfram Mathworld.

“Operation. Let A be a set. An operation on A is a function from a power of A into A. More precisely, given an ordinal number alpha, a function from A^alpha into A is an alpha-ary operation on A. If alpha=n is a finite ordinal, then the n-ary operation f is a finitary operation on A.” (Wolfram Mathworld [ http://mathworld.wolfram.com/Operation.html ]).

I take it a function could still be an operation in this narrow sense if the arity was 1, e.g. a mapping from the reals into or onto the reals. Here's another, broader definition I came across:

“Operation. 1. any procedure, such as addition, multiplication, set union, conjunction, etc., that generates a unique value [Unique to what? I’m presuming, by analogy with the definition of a function, unique to the input.] according to mechanistic rules from one or more numbers or values as arguments. 2. a function determined by such a procedure.” (Borowski, E.J. & Borwein, J.M.: Collins Dictionary of Mathematics). Would their Sense 1 be synonymous with "rule of assignment"?

For operator, I've found:

“Operator. 1a. any symbol used to indicate an OPERATION, such as the integral operator [integral symbol] and the differential operator [capital delta symbol]. b. the function determined by such an operation. 2. a MAPPING, such as a LINEAR OPERATOR” (Borowski, E.J. & Borwein, J.M.: Collins Dictionary of Mathematics). Following the trail of entries, they gives as synonymous: function, mapping and transformation, at least in some uses.

“Mapping or map. A function or transformation.” (Borowski, E.J. & Borwein, J.M.: Collins Dictionary of Mathematics).

“Map. The terms function and mapping are synonymous for map.” (Wolfram Mathworld [ http://mathworld.wolfram.com/Map.html ]).

“An operator [...] is [...] a mapping between two function spaces. If the range is on the real line or in the complex plane, the mapping is usually called a functional instead.” (Wolfram Mathworld: Operator [ http://mathworld.wolfram.com/Operator.html ]).

“Operator. The word operator can in principle be applied to any function. However, in practice it is most often applied to functions which operate on mathematical entities of higher complexity than real numbers, such as vectors, random variables, or mathematical expressions.” (Wikipedia). This one's especially helpful as it goes into some detail about the various naming practices, including notes to the effect that the term "operation" is also traditionally applied to a function that happens to have a traditional, commonly used symbol of its own, e.g. +.

For Davis & Snider, an operator is a name traditionally applied to "functions [...] that associate functions with functions" (An Introduction to Vector Analysis, sixth edition, 1991, p. 140). They (gently) deprecate the term operator for obscuring to beginners the broad application of the concept of a function.

 

[Tac-Tics]

An operation is a synonym for a function.

The word operation tends to have some additional connotations. One example would be the derivative, which is a function that acts on real functions. But that is a bit confusing, so we might call the derivative an operator. But it does the same exact thing as a function -- it takes an input (a real differentiable function) and assigns it a single output (the derivative of the input).

 

[Elucidus]

Yes, an n-tuple is an ordered n-tuple. However, I'm not sure all texts agree on whether there is order to the components.

You seem to have unearthed quite a bit of info on the usage of these terms already. Unfortunately the words are frequently used to mean different things and the same thing at different times.

You are correct, a function from Aⁿ to A is an operatION. In this sense the operatOR is the symbol used to represent it. So addition is an operation while its operator is "+".

The limit operator is "lim" while its arguments are "x," "a," and "f(x)" in \lim_{x\rightarrow a}f(x).

Operator in the functional sense is usually a function from a function space to itself. e.g.

D=(f\mapsto Df):\mathcal{F}\rightarrow\mathcal{F}

is the differential operator. (aka \frac{d}{dx}).

Functions and Operations must all be single valued. At one time long ago functions were permitted to be multivalued, but this lead to much ambiguity and the practice has since been wisely discontinued (some courses will occasionally dabble in it, but it is admittedly sloppy).

--Elucidus

 

[Rasalhague]

Yes, an n-tuple is an ordered n-tuple. However, I'm not sure all texts agree on whether there is order to the components.

I haven't yet encountered any text that doesn't make order a defining property of n-tuples, but I'm just a novice, so I'll keep that in mind. I guess a definition that didn't require order would make a set a special case of a tuple, rather than the other way around. Maybe... The definitions I've seen have taken the concept of a set as basic, and used this to define ordered pairs and n-tuples.

You seem to have unearthed quite a bit of info on the usage of these terms already. Unfortunately the words are frequently used to mean different things and the same thing at different times.

No surprises there then ;-)

Are operation and operator ever used as synonyms? Some of the definitions seem to imply that the senses coincide (when they mean function, map, mapping, transformation), but not all senses (as when operator means a symbol). I wonder if Borowski & Borwein's sense "procedure" for operation coincides with the "rule of assignment" component of a function.

Functions and Operations must all be single valued. At one time long ago functions were permitted to be multivalued, but this lead to much ambiguity and the practice has since been wisely discontinued (some courses will occasionally dabble in it, but it is admittedly sloppy).

Yes, I've met this issue. Would the preferable practice then be to refer to "multivalued functions" simply as relations?

One more definition of mapping, this from Carol Whitehead's Guide to Abstract Algebra: "Let X, Y be non-empty sets and alpha be a relation between X and Y. Then alpha is said to be a mapping if every element in the domain X is related to one and only one element in the codomain Y." There's also a note here that "not all the relations referred to as 'functions' in elementary calculus are mappings of R into R by our definition". Is this a reference to relations such as 1/x, for which at least one element in X is not related to an element in Y?

 

[Elucidus]

I haven't yet encountered any text that doesn't make order a defining property of n-tuples, but I'm just a novice, so I'll keep that in mind. I guess a definition that didn't require order would make a set a special case of a tuple, rather than the other way around. Maybe... The definitions I've seen have taken the concept of a set as basic, and used this to define ordered pairs and n-tuples.

I've seen n-tuples generalized to dennumerable coordinates, and I suspect you can push it to uncountable coordinates or worse non-totally ordered indexing sets. If you had a relation between ojects from disjoint sets A, B, and C but there was no required order between the sets, you could consider the triplet {a, b, c} which I guess could be interpreted as a 3-set, or in the case where overlap occured, a multiset. Et cetera..

Are operation and operator ever used as synonyms? Some of the definitions seem to imply that the senses coincide (when they mean function, map, mapping, transformation), but not all senses (as when operator means a symbol).

I think you've hit on the distinction. I see them used synonymously too.

I wonder if Borowski & Borwein's sense "procedure" for operation coincides with the "rule of assignment" component of a function.

A rule of assignment is a special type of procedure. Procedures are rules that do something, like draw a diagram, store information in the accumulator, print yesterday's email - as well as mathematical procedures. You could argue that the "output" is the result of what the procedure did, but this is taking a loose interpretation of codomain.

--Elucidus

 

[Rasalhague]

A rule of assignment is a special type of procedure. Procedures are rules that do something, like draw a diagram, store information in the accumulator, print yesterday's email - as well as mathematical procedures. You could argue that the "output" is the result of what the procedure did, but this is taking a loose interpretation of codomain.

What condition(s) must a procedure fulfil to be a rule of assignment? Is it that it has the suitable properties to be used as the rule component of a function, or could the rule component of any relation be called a rule of assignment?

 

[Elucidus]

What condition(s) must a procedure fulfil to be a rule of assignment? Is it that it has the suitable properties to be used as the rule component of a function, or could the rule component of any relation be called a rule of assignment?

This may be just my opinion, so if you find contradictory evidence elsewhere, I am happy to concede the point. I feel that a "rule of assignment" requires that it be single-valued. So it is an integral part of a function (note that a function is a kind of relation).

The rule associated with relations is probably best called a "rule of correspondence" or "rule of relation" but I cannot recall ever seeing these terms formally defined.

Usually you'll see a binary relation defined similarly to:

A binary relation R from a set A to a set B establishes for each a from A and b from B either that a is R-related to b (denoted aRb) or a is not R related to b (denoted a\text{ not}R \;b).

(That last expression usually involves an R-slash but I can't recall how to render that in TeX.)

Relations are most often associated with the set R=\{(a,b)\in A\times B:aRb\} so the "rule" gets kind of lost.

--Elucidus

 

[CRGreathouse]

A binary relation R from a set A to a set B establishes for each a from A and b from B either that a is R-related to b (denoted aRb) or a is not R related to b (denoted a\text{ not}R \;b).

(That last expression usually involves an R-slash but I can't recall how to render that in TeX.)

It should really be

a\not\operatorname{R}b

but it displays better (at least on the forums) with some negative space.

a\!\not\!\operatorname{R}b

a\!\not\!\operatorname{R}b