I just finished high school and started my journey with self-learning Maths. I have a question about Kuratowski Definition of Ordered Pairs: (a,b)={{a},{a,b}} I'm not quite sure what this means exactly even though I looked through multiple questions here and on quora. Is this (the right side) just a notation to indicate which element comes first? or there's actually some set-theory-logic behind it?
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3The left side is the notation, the right side is the definition. As a set, the object "$(a,b)$" is defined to be the set ${{a},{a,b}}$. One then proves that $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. You need a definition because sets do not have any intrinsic order. Under the usual definition of sets, ${a,b}={b,a}$. So if you want to have some way to say "$a$ first, $b$ second", you need to make some construction that achieves this. – Arturo Magidin Sep 30 '21 at 18:28
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@ArturoMagidin thanks for the elaboration now I have a clearer understanding of the definition. However, I still have questions (kinda the same as my original one ): 1) What is the actual logic behind writing the construction you mentioned like this: {{a}.{a,b}}. Is there underlying logic that makes this expression "ordered" or is it just a way to distinguish it from the regular-ordered set? 2) Can we apply this definition for ordered sets with more than 2 elements (If so how would it look like ) or is this definition applied only to pairs (like the name indicates). ? – Sajjad Emad Sep 30 '21 at 21:06
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The logic is: it works!. It has precisely the property you want, which is that $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. There are several different ways of making that happen, this is one of them. Once the construction has property you want, that is all you need. You can extend it to more than 2 elements, but one almost never does. Instead, you define functions using ordered pairs, and then define higher "tuples" using functions. – Arturo Magidin Sep 30 '21 at 21:11
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@ArturoMagidin I understand the logic behind the definition itself ((a,b)=(c,d) if and only if a=c and b=d) and I find it clear. What I'm struggling to understand is why we write the ordered pair like this : {{a,{a,b}}. I understand that the first set within the main set "{a}" indicates that (a) comes first and the second set "{a,b}" indicates the pair itself but my question is if there is a reason behind writing like this. The post you linked actually helped and I find the Hausdorff's way to be more clear but I'm still confused why we write the pair like this in Kuratowski's way {{a},{a,b"} – Sajjad Emad Sep 30 '21 at 21:30
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I realize that we cannot write the ordered pair in the regular set notation like this {a,b} because simply sets does not consider order and that we have to find a way within set theory to indicate order however I cant understand why we write it like this {{a},{a,b}}. How does indicates order exactly? (Sorry if I'm missing on something very obvious or something like this, I'm pretty new to this ) – Sajjad Emad Sep 30 '21 at 21:34
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As I already said twice , we define it that way (it's not just how we "write it", it's how we define it) because that set has the requisite property: that the set ${{a},{a,b}}$ is equal to the set ${{c},{c,d}}$ if and only if $a=c$ and $b=d$. That is what makes it "ordered": because $(a,b)$, $a$ first and $b$ second, is equal to $(b,a)$ ($b$ first, $a$ second) if and only $a=b$. In this set, which one goes in the singleton matters. – Arturo Magidin Sep 30 '21 at 21:39
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It's not that we are inventing a new way to write the ordered pair. It's that we need to explain what the ordered pair is as a set, because in Set Theory the only things that exist are sets. If you think that this is just a matter of notation (how "we write it"), then that's what the problem is. It's not. It's a definition. – Arturo Magidin Sep 30 '21 at 21:45
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@ArturoMagidin So is it possible to say that we're using sets to convey information about our ordered set and to describe it? Because in set theory we only have one tool (which is sets) and we have to use them somehow to indicate order (based on a definition like Kuratowski's). But in the end we write it {{a},{a,b}} to describe the ordered pair using sets? – Sajjad Emad Sep 30 '21 at 21:51
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There you go again. It's not about how "we write it". It's a DEFINITION. We write it as usual, $(a,b)$. For the fourth time: we need that symbol to specify some specific set. And whatever set it specifies, we want that set to have the property that the set corresponding to $(a,b)$ is equal to the set corresponding to $(c,d)$ if and only if $a=c$ and $b=d$. This particular definition has that property. That's what matters. As is discussed extensively in the duplicate I've indicated. – Arturo Magidin Sep 30 '21 at 21:56
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@ArturoMagidin I think I get it now. thanks for bearing with me and sorry for making you repeat yourself. I really appreciate the time and effort you took into this. I'll make sure to review it to deepen my understanding. I'd also appreciate if you could recommend beginner-friendly books for set theory. – Sajjad Emad Sep 30 '21 at 22:04
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Often in Mathematics, concepts arise from an intuition, but the formalisation is another story, often improved constantly through experience and the passage of time.
Without diving into the formalities, we just want a pair of elements with an order defined.
Why would we do that? Because it is interesting in the study of certain problems or areas of mathematics. It's not the same to walk north 3 metres and then 6 metres to the east, in contrast to walking 6 metres to the north and 3 metres to the east. There is notion of order, and we could think of $(3, 6)$.
However, mathematicians have been striving to build a solid ground for the building of mathematics, and in that effort, the set theory was born, the heart of all mathematics. So, it is natural that mathematicians try to define this thing called "$(a, b)$" using only the basics: the sets. So, in the end, this definition is just formality, you don't need it to understand what an ordered pair is.
Jumping back to the subject, notice that the set $\{3, 6\}$ is lacking when we try to represent this information. One would think a way to sort this out by defining a new set $\{\{3, 1\}, \{6, 2\}\}$. And for this case, it works. Eureka! Now we can convey the order using only sets! (This way of defining an ordered pair is called Hausdorff's definition). However, you might notice that it wouldn't work if we had the ordered pair $(2, 1) = \{\{2,1\},\{1, 2\}\} = \{1,2\}$.
Many definitions would fail with the passage of time, until some prevailed over the others, because they proved that they worked. One of such is Kuratowski's definition. How can we define order using ONLY sets and the elements given?
Kuratowski answers this question by defining $(a, b) = \{\{a\}, \{a, b\}\}$ . As you can see, the set $\{a, b\}$ conveys the information on what elements are present on the pair, and the set $\{a\}$ conveys the information on who is the first element.
Jiaze Zhang
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":=" is borrowed from computer science, and is pretty recent. It does not have a universally agreed upon meaning of "definition", the way, say, $=$ has a universally agreed upon mathematical meaning of "is equal to". Your parenthetical comment in the last paragraph reflects one usage, not a general usage as you imply. – Arturo Magidin Sep 30 '21 at 21:42
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@MarkSaving: In some computer languages it means nothing; and some mathematicians do not recognize it as a sensible mathematical symbol. Saying it is the mathematical symbol for "definition" is like saying that $<>$ is "the" mathematical symbol for inequality. It's not; in some context it is nonsense. Yes, I know this is the most common interpretation these days. – Arturo Magidin Sep 30 '21 at 21:47
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@MarkSaving: As for other symbols used to represent definitions, some use $\equiv$, some use $\stackrel{\triangle}{=}$, some use $\stackrel{\text{def}}{=}$. – Arturo Magidin Sep 30 '21 at 21:58
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I understand the intuition for ordered pairs as it's really simple however I find it interesting to deep dive into the pure mathematical definition. Thanks for your detailed answer. thanks to your answer and @ArturoMagidin 's comments now I have a better understanding of the concept. – Sajjad Emad Sep 30 '21 at 22:10
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@ArturoMagidin Indeed, however, it is also true that Kuratowski's is one of most widely accepted and used definitions of an ordered pair.
Also, it is true that giving the definition of ":=" is not entirely accurate. My mistake, I read his post too quickly and thought he used ":=", but he actually just used "=". – Jiaze Zhang Oct 01 '21 at 05:12