Can an ordered pair have identical elements?

[Stoney Pete]

Hi guys,

Here is a wacky question for you:

Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written as the set containing the ordered pair (x, x). But does such an ordered pair, with identical elements, make any sense? After all, order is everything in ordered pairs (and tuples generally), right? But how can one distinguish order when the members of the pair are the same?

I have read somewhere that the Kuratowski definition of an ordered pair (x, y) as {{x}, {x, y}} allows ordered pairs with identical elements, namely as follows: (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}. But how does having the set {{x}} tell us anything about order?

What are your thoughts on this? And do you know of any literature dealing with this issue? Thanks for your answers!

Stoney

 

[jambaugh]

If you eliminate ordered pairs with identical elements you couldn't define coordinates for the points on the line y=x. An ordered pair is specifically *not* a set so yes there's nothing odd about the pair having equal elements. Constructionists are fond of defining all mathematical structures in terms of sets to allow everything to have a common axiomatic footing. But there's no need to worry about such foundations when using mathematics. An ordered pair is an ordered pair. (a,b) = (x,y) if and only if a=x and b=y. Nothing more need be said except in specific applications.

 

[Stoney Pete]

I am kinda interested in axiomatic foundations, so for me the set-theoretic construction of ordered n-tuples is important...

When you say an ordered pair is specifically not a set, then that is in a sense wrong, since an ordered pair can be defined in set-theoretic terms (e.g. the Kuratowski definition mentioned in my first post).

And when you mention the linear function x=y you are missing my point. That linear function indeed gives a set of ordered pairs {(0,0), (1,1), (-1,-1),..., (n, n), (-n,-n)} where the elements in each pair are in a sense identical, but at the same time they specify different geometric values (the first one on the x axis, the second on the y axis). I mean a function where the input is in all respects identical to the output.

Also, I am not specifically talking about numerical functions but about functions in a more abstract logical manner as mappings from one set to another, no matter what is in the set. For example, causation can be seen as function mapping a set of causes to a set of effects. Now take the old philosophical idea of self-causation, where a thing (e.g. God) causes its own existence. You then have a function where input and output are identical. It is for such cases that I wonder whether the notion of ordered pairs still makes sense. I guess this is not so much mathematics but concerns rather formal logic of metaphysics... Nevertheless, it specifies a clear question in set theory and the theory of functions: can the elements of an ordered pair (or any n-tuple for that matter) be identical in all respects?

 

[Stoney Pete]

For a more mathematical example of what I have in mind, consider the function f:N→N where domain and range are the same set N. This function is a set including ordered pairs such as (1, 1), (2, 2) etc. How can we speak of ordered pairs in such cases, where there really is just one element mentioned twice?

 

[jambaugh]

I was not mentioning a function I was mentioning a geometric object, a line, and its coordinate representation, a set of ordered pairs.

I am not sure what is causing you confusion. Remember that the ordered pair is a collection, and not the objects in the collection. If you prefer you may think of the "slots" in the pair as representing a pair of variables. You can have distinct variables x and y that may happen to have equal values (x = 3 and y= 3 at the same time).

I would avoid thinking in terms of functions explicitly as the ordered pair concept is more primitive (as in gets defined first).

Another point: in your comment you seemed to imply you were thinking of the order in the ordered pair as having to do with some intrinsic ordering of the elements. This is not the case. The pair ordering is not a property of the entries, the entries are properties of the pair object. (one is the property of "first value", the other is the value of the "second" property.)

A pair is just a list of length 2 (with the natural generalization of triples, quadruples, ... n-tuples.) The ordering is simply a matter of the pair NOT being a two element set.
Where you ask:

How can we speak of ordered pairs in such cases, where there really is just one element mentioned twice?

The answer is that we can mention the same object twice, "potato, potato". And a pair is simply the mathematical semantics of making two references to an object or objects. The two references may refer to the same object or different objects because there's no proscription denying that possibility in the definition of the pair.

Its just like the words we use in alphabetical written language. There's no problem allowing letters to repeat or occur more than once in a word. Think of an ordered pair as a "two letter" mathematical "word". mm?

Once you understand this intent in the definition of an ordered pair then you can choose your favorite way to axiomatically construct it from other objects, be they sets, or categories, or functions, or groups.

 

[Stephen Tashi]

I have read somewhere that the Kuratowski definition of an ordered pair (x, y) as {{x}, {x, y}} allows ordered pairs with identical elements, namely as follows: (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}. But how does having the set {{x}} tell us anything about order?

I am kinda interested in axiomatic foundations,

Pity you! $\$ But I see your point - you aren't in doubt about the intuitive meaning of "ordered pair".

However, you aren't asking a precise question. Your question asks how a particular set "tells us anything about order". This implies you have a definition of "order" in mind that the set should tell us something about. So we don't have a precise question until you offer a definition of what it means to "tell something about order".

If we take for granted that the natural numbers are defined or that ordinal numbers are defined, you could ask how the Kuratowski definition answers some question involving those mathematical concepts - i.e. Does the Kuratowski definition tell us if an element is the "first" element of an ordered pair ? However, what definition of order are we using to create the definitions of the natural numbers and ordinal numbers? - e.g. if we ask a question about a "first" thing, what definition of "first" are we using in our question?

The Kuratowski definition you quoted doesn't mention the terms "first member of the ordered pair " and "second member of the ordered pair", so it's fair to say the Kuratowski definition tells us nothing about the meaning of those terms. The game of axiomatics is to begin with certain mathematical concepts and to use those concepts to define more mathematical concepts. (It's a game that can be played in more than one way.) We want our precisely constructed creations to match our intuitive ("Platonic") notions of familiar mathematical objects. So, from the point of view of axiomatics the relevant question is not whether the Kuratowski definition contains within it the definitions of "first" and "second". The relevant question is whether we can use the Kuratowski definition and other already-defined concepts to construct a definition of "first member " and "second member" that matches our intuitive idea of those concepts.

Can we do that? ( I don't know what Kuratowski did, but I think we can accomplish that task.)

 

[FactChecker]

The mapping:
(x,y) ↔ {{x}, {x,y}} if x ≠ y
(x,x) ↔ {{x}}

Seems well defined and usable for defining ordered pairs. I don't immediately see why one would want to do that, but that might just show my lack of imagination.

 

[Stephen Tashi]

The mapping:
(x,y) ↔ {{x}, {x,y}} if x ≠ y
(x,x) ↔ {{x}}

Seems well defined and usable for defining ordered pairs.

From an axiomatic point of view, we need the definition of "ordered pair" before we can construct the usual definition for a "mapping" as a "set of ordered pairs such that ...".

 

[FactChecker]

From an axiomatic point of view, we need the definition of "ordered pair" before we can construct the usual definition for a "mapping" as a "set of ordered pairs such that ...".

Right. I guess I should have said it is an association between two alternative (hopefully equivalent) definitions.

 

[Stephen Tashi]

An interesting technicality in Kuratowski definition of ordered pair is the question of how much human perception of notation is allowed to play a role. If we phrase the definition in the form:

"The ordered pair (a,b) is defined to be the set {{a},{a,b}}"

then we have assumed human perception distinguishes "(a,b)" from "(b,a)" and thus we assume there is a perceived property of order ( left-to-right) that is utilized in making the definition, but not explicitly explained by the definition itself.

From the purely axiomatic point of view, the definition of an ordered pair would be better written in the form like:

"P is an ordered pair" means that ...

so that no undefined notion of "first" or "leftmost" would be assumed.

I wonder how Kuratowski wrote the definition in his original papers.

 

[Logical Dog]

yes it can. :)

in a function, the ordered pair definition usually means that the element on the left is of the domain, and that on the right is the co domain.
that is why you see usually a function as an ordered pair defined as

N x R or similar, it is said to be a binary relation

the cartesian plane R^2 is an example of an ordered pairing where same elements exist,
functions as you said, and even relations

also as far as I know an ordered pair is an element of a set, and thus, set properties and operations on it apply

the number of elements inside the ordered pair also reflect the number of sets which have been combined IN ORDER to produce it.

So {(a, b , c, d) | a in X, b in Y, c in Z, d in Q}.

the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b.
ordered pairs are also used accodring to book of proof by R HAMMACK. to define relations such as < > =.

So the = relation on the cartesian plane is the set of all ordered pairs where (a,a). google reflexive relation.

 

[FactChecker]

also as far as I know an ordered pair is also a set, and thus, set properties and operations on it apply

You are correct. Kuratowski's definition is valid. It's just that, as you said, {{a}, {a,b}} = {{a}, {b,a}} represents (a,b) and {{a}, {a,a}} = {{a}} represents (a,a).

 

[Logical Dog]

You are correct. Kuratowski's definition is valid. It's just that, as you said, {{a}, {a,b}} = {{a}, {b,a}} represents (a,b) and {{a}, {a,a}} = {{a}} represents (a,a).

sorry, I meant the ordered pair (a,b) that is an element of some larger set of ordered pairs, meaning the set of ordered pairs is a set, and (a,b) its element. BUT that's what you understood.

 

[Logical Dog]

IF YOU are having trouble composing ordered pairs, it is easier to LIST out all elements horizontally, and LIST out the other sets elements Vertically, and combine them...ALSO for a function the domain is always the horizontal axis, the co domain the vertical.

{1, 2, 3} x { a, b , 1}
-----1 2 3
a
b
1

edit: HOLD on..I don't understand how (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}..??

for the set {{x}, {x, x}} you get this ordered pair (assuming the set is "mutliplied" by itself):
(x,x), (x,x,x) and other ugly stuff? but i agree that {x} x {x} = (x,x)

are you taking the power set of (x, x)? bUT (X,x) IS NOT a set. it is a list that is part of a set containing other lists.

power set of {(x,x)} = {{}, {(x,x)}}

 

[FactChecker]

edit: HOLD on..I don't understand how (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}..??

Sorry, in post 12, I was thinking that you were the OP. See the discussion in post 1. This is just his original definition followed by the basic set property that repeated elements can be ignored.