Odds and ends

[Dale10101]

I couldn't decide where to ask this question.

Is addition on the set of real numbers a binary relation or a ternary relation?


If I look at it as an statement with three places/operands, (x, y, z) where x + y = z then, Addition(x, y, z), as a relation, is a subset of the Cartesian product X x Y x Z and hence ternary ... I would think.

On the other hand,

if I look at as ( (x,y),z) then I would say it is a binary relation, a subset of the Cartesian product, (X x Y) x Z.

Can it be that addition is both a "binary operation" and a "ternary relation".

I am getting hung up on what it is called when you take a relation with three places (x, y, z) and divide it into a set consisting of a domain element (set of departure) and a co-domain element (set of destination), say ( (x,y), z).

Would that be the point where you go from calling Addition a "relation" to an "operation"?

Analogous to the above questions, is the unary operation of taking a square root, a binary relation?

For whatever reason I am bugged by understanding how to name and classify math concepts as well as how to "do math".

 

[Ishuda]

I couldn't decide where to ask this question.

Is addition on the set of real numbers a binary relation or a ternary relation?


If I look at it as an statement with three places/operands, (x, y, z) where x + y = z then, Addition(x, y, z), as a relation, is a subset of the Cartesian product X x Y x Z and hence ternary ... I would think.

On the other hand,

if I look at as ( (x,y),z) then I would say it is a binary relation, a subset of the Cartesian product, (X x Y) x Z.

Can it be that addition is both a "binary operation" and a "ternary relation".

I am getting hung up on what it is called when you take a relation with three places (x, y, z) and divide it into a set consisting of a domain element (set of departure) and a co-domain element (set of destination), say ( (x,y), z).

Would that be the point where you go from calling Addition a "relation" to an "operation"?

Analogous to the above questions, is the unary operation of taking a square root, a binary relation?

For whatever reason I am bugged by understanding how to name and classify math concepts as well as how to "do math".

You might want to look at
https://en.wikipedia.org/wiki/Arity

In a formal sense an operation's (function's) 'arity' is associated with the number of elements needed to perform that operation. That is, starting at the most basic, addition requires two elements to perform the addition operation so addition is a binary operation/function. Negation, square root, cosine, ... would be unary operations/functions, etc.

 

[Dale10101]

Thanks

You might want to look at
https://en.wikipedia.org/wiki/Arity

In a formal sense an operation's (function's) 'arity' is associated with the number of elements needed to perform that operation. That is, starting at the most basic, addition requires two elements to perform the addition operation so addition is a binary operation/function. Negation, square root, cosine, ... would be unary operations/functions, etc.

Thanks for your quote and the link, I think it has clarified things, especially the meaning of "arity" ... so what I take away is:

Given sets [A, B, C, D] the corresponding Cartesian Product is A x B x C x D, all possible quaternaries (a, b,c, d)

A relation L on that Cartesian Product is specified as a subset of A x B x C x D, all possible quaternaries (a, b,c, d), or fewer.

One can next add additional information to create a function/operation F on L by defining a domain and co-domain for F as exclusive proper subsets of the domain of L.

In this case, most typically one would declare a domain set as A x B x C with a co-domain of D. The result would be a ternary function
F₁ : (AxBxC) -> D

I am supposing one could also create a Binary function on L as F₂​: (AxB) ->(C,D),

or even a unary function on L as F₃: A -> (B x C x D). Naturally such functions would be expressions whose set of correspondences between it's domain and co-domain would be functional.

I also suppose that the term "relation" is ambiguous, or context dependent, in that R: (A x B x C) -> D could define a "relation" on the "RELATION" L in that some particular correspondences R(a,b,c) -> D might not be unique.

Finally it seems to me that the difference between a "function" and an "operation" is simply that, in general, the term operation is reserved for naming the lowest level functions on a domain, like addition, multiplication, exponentiation, etc, or as a verb describing the process of executing a function.

Hopefully I am getting this straight, of course, it might me that the parenthetical element of the statement from the wiki link,


"The arity of a relation (or predicate) is the dimension of the domain in the corresponding Cartesian product. (A function of arity n thus has arity n+1 considered as a relation.)"

must be strictly interpreted meaning that my suppositions about creating binary and unary functions on L are wrong. (?)

(Sorry if I beating a dead horse, just want to make sure its not sleeping. Hmmm, I wonder if there is an anthology of politically incorrect expressions from bygone eras.)