Understanding Ordered Pairs to Notation and Definitions

[matt grime]

by "SET" do you mean the language of set theory?

no, the category of sets.

does "construct" mean something like "represent symbolically"?

no, it means construct, make, produce a set with the required properties. a lot of proofs in maths show existence wiithout ever actually demonstrating a single example. it is easy to show that almost all numbers are transcendental, and but very hard to actually find many of them.

please define "projection".

exactly what you think it means: the map from AxB to A sending (a,b) to a (and the one defined on B)

"that is universal with this property": it's not clear to me what "that" refers to, and which property.

having these projections. google for category theory and product, because it is best explained by a picture thaht i can't draw here.

finally (and I don't mean this at all in a disparaging way) does mean that you (set-theorists en masse)

I'm definitely not a set theorist; i know next to nothing about set theory; can't stand it in fact

can't define these concepts in words but only by written symbols?

we could use words, we don't since symbols are universal. personally i tihnk set theory over relies on symbols, but if you dont' like symbolic definitions then maths is going to get quite bad since few books bother with wordy descriptions, possibly at their own expense

When you discuss these things with your colleagues, do you always carry paper along so you can draw these little pictures to flash at each other? (Sorry, couldn't resist that.)

the language used when discussing maths informally and the formal language of books and definitions are not the same.

 

[gnome]

This is like trying to catch a greased pig.
I can appreciate the utility of symbolic representations, but sometimes words are useful too. I can't understand why it is so difficult to get you guys to attach words to this one:
{{a},{a,b}}
You're standing in front of your class, and there's a blind guy named Charlie in the class.
You say: "An ordered pair is defined as ... " and here your voice trails off as you write on the blackboard

{{a},{a,b}}

"this!" you triumphantly pronounce. "Sorry, Charlie, you're out of luck."
Are you going to stick to:
An ordered pair is defined as left bracket left bracket a right bracket comma left bracket a comma b right bracket right bracket
and that's it? Can't you come up with some words to help Charlie out?
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I wonder why Enderton felt it appropriate to present this definition, with no further discussion, in a "review" chapter (in an introductory logic text) that begins

We assume the reader already has some familiarity with normal everyday set-theoretic apparatus. Nonetheless, we give here a brief summary ...

We don't seem to be dealing in this thread with "normal, everyday" (which I take to be what Matt calls "naive") anything. Well, maybe normal, everyday for set-theorists. Enderton's book is clearly not aimed at set-theorists. We deal with ("naive", I guess) set concepts constantly, in computer science and in many other disciplines, without losing any sleep over whether or not the set of all sets is a set.
I will try to find time to look at Enderton's "Elements of Set Theory". Maybe he sheds more light on this definition there. And I clearly have to look into category theory, which in my ignorance I thought was a branch of set theory. I stand corrected. Maybe you can recommend something for me to read. Is there such a thing as "Categories for Non-mathematicians"? (I'm not, nor have I ever been, a math major -- just trying to get a reasonably clear idea of what you guys find so irresistable.)

 

[matt grime]

Naive is its name, it is not an invention of mine, it is the set theory that is used and used without problem for 99.99999999% of maths.

You want it in words? OK, we define set of ordered pairs of AxB to be the set constructed as follows: we take all the single elements sets whose element is in A, and the collection of all pairs of elements which contain exactly one element of A and one of B, then we take the subset of the combined collection of all these sets where the element of the singeleton set is an element of the pair.Happy?

I can define it in words more simply but not in words that are purely set theoretic since sets do not have orderings!

 

[Hurkyl]

Hrm...

If you were to ask "What are the real numbers?" instead of "What is an ordered pair?", would you be looking for the "The real numbers are a complete, ordered field" answer, or the "Real numbers are equivalence classes of Cauchy sequences" answer?

 

[mathwonk]

an ordered pair is a pair of things, and one of them is designated as the first one.

if you do not believe such things exist, i feel sorry for you.


If you really want to understand how the set theorists have labored to define such an animal within set theory, I can understand your predicament, having once been there myself, as young child.

But by now I have lost all interest in this exercise, as I have never used the concept in the succeeding 42 years.


So have most people. This explains the reluctance to give lengthy answers to such questions, when many other more interesting mathematical questions abound.

I.e. if this were considered cutting edge mathematics, we would have all become real estate agents long ago.

 

[gnome]

Matt: yes that makes me feel happy. Dumb, but happy, thank you very much. Actually, you could have stopped at "we define set[/color] of ordered pairs of AxB[/color] to be" because that in itself made it all clear. You see, I was not reading it as the definition of a set; I was trying to read the definition of a general concept: "An ordered pair is ..." which of course made no sense, and I needed you to hit me over the head with those words to make me see what you meant.

Hurkyl: thank you too, but I know better than to ask that question. Seriously, it would take more time than I can afford in order to really understand either of those answers. For the time being, I'll sleep very well with a 1-to-1 correspondence to the points on an infinite line.

mathwonk: you misunderstand my motivation in all this nonsense. This was not a debate. Until three days ago I was very satisfied with "an ordered pair is a pair of things, and one of them is designated as the first one" (and tomorrow I will again be satisfied with that). Unfortunately, I came across that {{a},{a,b}} definition in Enderton's "Mathematical Introduction to Logic" and foolishly thought that if it was important enough for him to put there, it was important enough for me to try to understand. Hurkyl and Matt were kind enough to waste their time helping me.

 

[Hurkyl]

Hurkyl: thank you too, but I know better than to ask that question. Seriously, it would take more time than I can afford in order to really understand either of those answers.

I only meant it as an analogy. I was trying to give a way for you to clarify what kind of answer you want, by analogy with this other case!

Unfortunately, I came across that {{a},{a,b}} definition in Enderton's "Mathematical Introduction to Logic" and foolishly thought that if it was important enough for him to put there, it was important enough for me to try to understand.

It is!

I think you're trying to get the wrong lesson from it, though: I suspect that (aside from completeness's sake) the point of this is that it's an example of the process of modelling a concept in some theory.



What happens in practice is that a budding mathematician is told "This is how we can define an ordered pair in the language of sets. Now let's just use the notation (a, b) and never speak of this again." (I think we've hinted at this, but I don't remember if we've said it explicitly)

 

[mathwonk]

i know that gnome. I was answering your persistent questions about why people were answering your questions with questions, and generally having to be begged to stay on topic. it just isn't a topic that seems important after grasping it.
nonetheless, it was rude of me, and I apologize.

Like you, I encountered this definition in a book, and assumed it must have some importance since it was given there (Kelley's Continental classroom algebra book? in 1960?).

I remember thinking only slightly sceptically at the time, " so this is really the definition of an ordered pair? Do I need to know this?"

The answer is no, and I think the only reason this was included in the book, was that someone decided at the time of the set theory hysteria that's wept the math ed world in 1960, that it could be done, and showed off by doing it.

However I think Kelley may have honestly remarked there that it could be done that way if desired, implying that ti had limited significance.

I did however work the exercise that {{x}, {x,y}} = {{z},{z,w}} if and only if x=z and y = w.

assume for example that x = y. then the first set contains only {x}. thus the second set which equals it, must contain only one set too, so we must have z = w also. then since {{x}} = {{z}}, we must have x = z.

now suppose x is different from y...

interesting huh?...:zzz:

 

[matt grime]

I can't wait until you come across what a set theorist thinks the natural numbers really are. If you think the ordered pair stuff is unnecessarily fussy wait until you see this.

 

[mathwonk]

yeah, i just reviewed a book on this stuff supposedly written for young math majors learning proofs. the definition of multiplication of integers in there was really off putting.

those guys must have been smoking crack. then the publishers advertised it by saying it was user friendly, and "written in a highly understandable style".

what a joke.

 

[HallsofIvy]

an ordered pair is a pair of things, and one of them is designated as the first one.

The problem with that is that you need to define what "first one" means- in other words, to define "ordered pair" you'd better first define an order!

The definition: "An ordered pair, (a,b), is a set of the form {{a},{a,b}}" avoids that- it doesn't talk about "first" or "second", "left" or "right", "top" or "bottom". It asserts that there are two objects (a and b) and that
(a,b)= {{a},{a,b}} is different from (b,a)= {{b},{a,b}}. You are then free to assume whatever "order" you want. That's all that "ordered pairs" have to say.

 

[Sir_Deenicus]

yeah, i just reviewed a book on this stuff supposedly written for young math majors learning proofs. the definition of multiplication of integers in there was really off putting.
those guys must have been smoking crack. then the publishers advertised it by saying it was user friendly, and "written in a highly understandable style".
what a joke.

Dont worry, the future is not doomed. Most of us with real interest tend to stay away from such trashy books and even have copies of books from those passed away 100 years gone!

 

[Hurkyl]

The problem with that is that you need to define what "first one" means- in other words, to define "ordered pair" you'd better first define an order!

Or, you could take the interpretation that "first" is merely a name for one of the elements, and has nothing to do with any notion of an ordering.

 

[CrankFan]

Unfortunately, I came across that {{a},{a,b}} definition in Enderton's "Mathematical Introduction to Logic" and foolishly thought that if it was important enough for him to put there, it was important enough for me to try to understand. Hurkyl and Matt were kind enough to waste their time helping me.

The concept of translating ordinary mathematical statements into a formal language is relevant to mathematical logic. Enderton could have omitted that brief exercise but if a reader struggles to see how, in principal, the concept of an ordered pair can be expressed set theoretically, then the reader is probably going to struggle with things presented later in the book.

Set theorists & Mathematical Logicians don't study formal theories because they provide a justification or because knowledge of them is required for other branches of mathematics. They study formal theories because they find it interesting. Same reason why number theorists study integers. If you're not interested in formal theories, don't tie yourself up in knots about it -- just stick to subjects that you enjoy.

 

[gnome]

I'm glad I was able to keep you all entertained these past few days.

I can't wait until you come across what a set theorist thinks the natural numbers really are. If you think the ordered pair stuff is unnecessarily fussy wait until you see this.

You'll be the first to know.

They study formal theories because they find it interesting. Same reason why number theorists study integers. If you're not interested in formal theories, don't tie yourself up in knots about it -- just stick to subjects that you enjoy.

As I said, there's no danger that I'll be crowding the ranks of the set-theorists. On the other hand, I don't consider this to have been a waste of time.

For example, this is the first time I've come across this idea of construction of mathemetical objects. Seems like a surprisingly "empirical" activity for mathematicians to be involved in. Personally, I'd rather tinker around under the hood of my TR-7, but I can see how you could find that interesting.

And, all kidding aside, I think the question of how to express the idea of ordering is fascinating. We all have this concept of order in our minds, but it takes so many forms depending on the particular subject to which it's being applied that it seems like a fairly daunting challenge for AI and one certainly worthy of attention.

 

[matt grime]

For example, this is the first time I've come across this idea of construction of mathemetical objects. Seems like a surprisingly "empirical" activity for mathematicians to be involved in.

It is not empirical at all. Think of it like this:

We take a theory of some type, written in some formal language, if you want to think of it like that. Then we make some formal statement about the objects that is not written in the original formal language. The question is if we can write this statement in that original formal language.

If you're interested in the Category theoretic viewpoint then it is occasionally useful to know when the construction we define is in 'the same universe' (which has a strict set theoretic meaning) as the things we first start from.


As a concrete example, consider the set of rational numbers, this is closed under finite sums of rational numbers (as in sums x_1+x_2+...+x_n a finite number of rationals added up). We can define a space of potentially infinite sums, whereby the infinite sums converge if and only if the sequence of finite partial sums is cauchy. This space cannot be the rational numbers, and is indeed the reals. Now suppose we ask the same question again whereby we allow for all infinite sums of reals whose partial sums are cauchy, do we get anything new? No, since the reals are complete.

Another example, that is easier to explain: consider the real numbers, and then ask if we can find all roots of all real coefficient polynomials in the reals. We can't but we can add in i and get the complex numbers. Is C then algebraically closed? Yes, it is.


Now, set theoretically, consider the class of all finite sets, is this closed under arbitrary unions? No, it is not, indeed there is a strict hierarchy, indexed by the cardinals, that indicate the sets one can make after taking different kinds of unions.

 

[gnome]

Just kidding.

I do get your point, and I intend to read more about category theory but I'm juggling too many balls to get involved in that right now.

Another day. Thanks again.

 

[jim mcnamara]

The set theoretic definition of a tuple uses the ordered pair as it's starting point. The result is that an n-tuple is an ordered set of n elements (or objects). -- Under this definition.

However, a tuple in database theory is usually defined as a finite function that maps fields in a relation(database table for instance) to data, and is not necessarily an ordered set of n elements. Another name for it is a "row" in a resultset or a row in a relation.

 

[mathwonk]

halls, surely you jest.

 

[mathwonk]

Aha! Halls you are right! Forgive me, but now I see for the first time, there really is no way to decide which element is "first"
.
For example when a = b, we have the proposed definition of the "ordered pair" (a,b) as the set {{a},{a,a}}. Now since {a} = {a,a}, this is just
{{a},{a}}, which of course is just {{a}}.
Thus there is no first copy of the element a.
Now this is clearly nonsense.
I.e. suppose we define an ordered singleton as a set of form {{a}}.
Then there is a non trivial intersection between the colection of ordered singletons and the collection of ordered pairs!
I.e. then when given the object {{a}}, we have no way of knowing whether it is an ordered singleton or an ordered pair.
This is obviously ridiculous and only someone who never planned to use them, would define them this way.
For instance with this definition, one could not tell the difference between the point (a,a) of the diagonal in the euclidean plane, and the point a on the real line.
So in fact the definition proposed for the ordered pair (a,b), namely {{a},{a,b}}, is a bit odd.
So unless I have made another of my many flagrant errors of thought (which is highly likely, given the huge number of intelligent people who have dutifully repeated this definition in their books), this silly definition is not only unimportant, but has odd properties.
e.g. in R^3, how should we define an ordered triple? perhaps we set (a,b,c) = the ordered pair (a,(b,c))? i.e. {{a},{{a},{b,c}}}.
Oho! We get the following: the "right" silly definition in this tradition, for an ordered singleton is of course not {{a}} but {a}.
then the ordered pair (a,a) is just {{a},{a,a}} = {{a}}.
and the ordered triple (a,a,a) is hmmm... {{a}, {{a}}}, so now R is disjoint from R^2, etc...
wheee... this kind of trivial nonsense is fun, until you wake up and realize you have more important things to do. It is kind of addictive though, like watching television. I fear it is equally harmful, as I am losing time I could be spending thinking about something deeper, like differential equations, or skew symmetric line bundles on curves.

 

[Hurkyl]

I've always wondered about the existence of strange and unusual statements one might be able to prove through these standard models. For example:

3 = 1 U (0, 1)

Because:
0 = {}
1 = {0}
2 = {0, 1}
(0, 1) = {{0}, {0, 1}} = {1, 2}
3 = {0, 1, 2}
1 U (0, 1) = {0} U {1, 2} = {0, 1, 2} = 3


At one time, I satisfied myself that none of this is a worry, if you rigorously `type' things. (I.E. don't try to ask about the equality of things of different `types')

 

[Sir_Deenicus]

Aha! Halls you are right! Forgive me, but now I see for the first time, there really is no way to decide which element is "first"
.
For example when a = b, we have the proposed definition of the "ordered pair" (a,b) as the set {{a},{a,a}}. Now since {a} = {a,a}, this is just
{{a},{a}}, which of course is just {{a}}.
Thus there is no first copy of the element a.
Now this is clearly nonsense.
I.e. suppose we define an ordered singleton as a set of form {{a}}.
Then there is a non trivial intersection between the colection of ordered singletons and the collection of ordered pairs!
I.e. then when given the object {{a}}, we have no way of knowing whether it is an ordered singleton or an ordered pair.
This is obviously ridiculous and only someone who never planned to use them, would define them this way.
For instance with this definition, one could not tell the difference between the point (a,a) of the diagonal in the euclidean plane, and the point a on the real line.

I think the definition works, and I do not think it is meaningful to infer backwards. I think <x,y> => {{x}, {x,y}}. That when x = y, we have a singleton means that since all parameters are equal the choice of first is ambigious or trivial. Because {{x},{x,y}} is a defintion and a convention its meaning can only be inferred by context (for example with knowledge of what space you are working on). {x} is drawn out first only to state which element is first in purely set theoretic terms in order to define some concept of order.

 

[mathwonk]

note however the difference between this and the more usual definition of a set of ordered pairs as a map with domain {1,2}. then the ordered pair (a,b) is a function f with f(1) = a and f(2) = b, so there are two "copies" of a, one taken first and one taken second.
to say that when both elements of the pair are equal, one does not need to designate one as first is different from saying there are two elements, one first one second, and both are equal. that is all i ma saying: i n ever noticed thius distinction before, and it violates my sense of...
hey what am I doing! I am going on with this trivial topioc. also if you read the remainder of my previous post you will see i have already pointed out the answer to my own paradox.
notice that in most cases a math talk can never be too elementary, the more elementary it is the mroe people understand and the mroe they respond. similarly a thread can never be too trivial. the moreso, the mroe resposnes it generates,
perhaps if i ask why 1+1 = 2, we will have another deathless thread.