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I have been looking at some of the posts here about expressing the ordered pair (a,b) in terms of sets as:

$$ (a,b) = \{\{a\},\{a,b\}\} $$

By definition, (a,b) is a set, so it should not 'care' about the order in which they are presented, so (a,b) should be equal to (b,a). I understand that the main property of ordered sets is that

(a,b) = (c,d) ,if and only if, a = c and b = d.

So

$$ (a,b) = \{\{a\},\{a,b\}\} = \{\{a, b\},\{a\}\} = \{\{c, d\},\{c\}\} = \{\{c\},\{c, d\}\} $$

How can I check the property of the set? Since they are sets I do not see how their order would now matter if it should not? If you can refer me to other sources, I have started a discrete book of math from scratch and they just showed the set definition without really explaining how it worked differently. Please keep your answer as simple as possible, I started this book as a recommendation from a math major to start learning math 'from scratch'.

What if we say $$ (a,b) = \{\{a\},\{b\}\} $$ wouldn't this obey the property such that if (a,b) = (c, d) then a= c and b =d?

RMS
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    As you observed, $(a,b)\neq(b,a)$. That's allowed because $(a,b)\neq{a,b}$. The order of elements of a set does not matter. – Rushabh Mehta May 04 '22 at 22:14
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    Have you read any of the related posts on this site? Such as: https://math.stackexchange.com/questions/62908/how-can-an-ordered-pair-be-expressed-as-a-set – Joe May 04 '22 at 22:17
  • There is a difference between 'learning math from scratch' and 'studying how fairly elementary mathematical objects can be given a set-theoretic basis'. Which is it that you want to do? I mean, I could give you a book that requires you to program in sequences of 1's and 0's ... but would that really be the best way to learn programming? – Bram28 May 04 '22 at 22:21
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    $(a,b)$ is a set, but it is a set that represents the ordered pair, so order does matter. The order of ${{a},{a,b}}$ doesn’t matter, but the order of $a$ And $b$ matters. – Thomas Andrews May 04 '22 at 22:22
  • The whole point is to find a way to encode an ordering of a pair using unordered sets. – Thomas Andrews May 04 '22 at 22:23
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    "By definition, (a,b) is a set, so it should not 'care' about the order in which they are presented" - this is correct. But it doesn't tell you $(a,b) = (b,a)$. It tells you that ${{a},{a,b}} = {{a,b}, {a}}$. Both ${{a},{a,b}}$ and ${{a,b}, {a}}$ are equal to $(a,b)$ and neither are equal to $(b,a)$ (unless $a=b$). – Jair Taylor May 04 '22 at 22:24
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    The idea is that you can tell from $S = {{a},{a,b}}$ which of $a$ and $b$ come "first" in the ordered pair $(a,b)$ because you can pick out the element of $S$ that has cardinality $1.$ For example, if I give you the set $S={{2,3}, {2}}$, you can see $S=(2,3)$ because $s=2$ is the only $s$ so that ${s} \in S$ and so $2$ must come first in the ordered pair. It's not equal to $(3,2)$ because ${3} \notin S$. – Jair Taylor May 04 '22 at 22:31
  • Hey guys! I really appreciate your constructive comments. They have help me understand and correct the misunderstandings I had. Other posts on the stack were useful but hard for me to understand them fully. I really appreciate your help. – RMS May 04 '22 at 23:24

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