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In the book "mathematical logic", it is said that $(a,b)$ is an abbreviation for $\{ \{a,a\},\{a,b \}\}$.

I don't understand this, since firstly, $\{a,a\}=\{a\}$, and secondly, even if this weren't the case, how does $\{ \{a,a\},\{a,b \}\}$ capture the ordered relation of $(a,b)$?

i.e. how do we see a tuple as a set?

user56834
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    See [Ordered pair](https://en.wikipedia.org/wiki/Ordered_pair#Informal_and_formal_definitions). Correct: $\{ a,a \}$ is the same of $\{ a \}$; and the def "works" because it capture the only relevant property of ordered pairs: $(a,b)=(c,d) \text { iff } a=c \text { and } b=d$. – Mauro ALLEGRANZA Jan 23 '18 at 13:22
  • $(a,a) = \{a,\{a, a\}\} = \{a, \{a\}\}$. $\{a,\{a, b\}\} = \{a, \{a, c\}\} \implies c = b$ – Good Morning Captain Jan 23 '18 at 13:22

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$f(a,b) = \{\{a\}, \{a,b\}\}$ captures ordering in the following way: $a$ is contained in the both elements of $f(a,b)$ and $b$ is contained in only one element.

Artur Riazanov
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  • You need to argue for the case that $a=b$ as well. Also, that indeed $(a,b)=(c,d)$ implies that $a=c$ and $b=d$. Your argument is essentially correct, but far from being sufficient as an actual answer to this question. – Asaf Karagila Jan 23 '18 at 14:11