Ordered Pair Formal Definition

Question

I understand what an ordered pair is, but I'm having trouble the formal Kuratowski definition which says that $\langle a,b \rangle = \{\{a\},\{a,b\}\}$. What about this definition imposes order on the pair?

Answer 1

notice :< a,b > = {{a},{a,b}} but < b,a > = {{b},{a,b}} the first element of these sets are different

Answer 2

The Kuratoiwski definition intends to enforce the one basic notion of an ordered pair, that is $$\langle a,b\rangle=\langle c,d\rangle\iff a=c\land b=d.$$ While one direction is trivial, note that $$\begin{align}&\langle a,b\rangle=\langle c,d\rangle\\ \implies&\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}\\ \implies&\{a\}\in\{\{c\},\{c,d\}\}\\ \implies&\{a\}=\{c\}\lor\{a\}=\{c,d\}\\ \implies&a=c\lor a=c=d\\ \implies&a=c\\ \end{align}$$ and then $$\begin{align}&\langle a,b\rangle=\langle a,d\rangle\\ \implies&\{\{a\},\{a,b\}\}=\{\{a\},\{a,d\}\}\\ \implies&\{a,b\}\in\{\{a\},\{a,d\}\}\\ \implies&\{a,b\}=\{a\}\lor \{a,b\}=\{a,d\}\\ \implies& b\in\{a\}\lor b\in\{a,d\}\\ \implies & b=a\lor b=d \end{align}$$ and by symmetry also $d=a\lor d=b$. Combined, this yields $(b=a\land d=a)\lor b=d$, i.e. $b=d$. In summary, $$\langle a,b\rangle=\langle c,d\rangle\implies a=c\land b=d.$$

Answer 3

It implies $\langle b,a\rangle=\{{\{{b\}},\{{a,b\}}\}}$, so $\langle a,b\rangle\neq\langle b,a\rangle$.