Ordered Pair Formal Definition 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?
 A: 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,
$$\langle a,b\rangle=\langle c,d\rangle\implies a=c\land b=d.$$
A: Notice: $\langle a,b \rangle = \big\{\{a\},\{a,b\}\big\}$, but  $\langle b,a \rangle = \big\{\{b\},\{a,b\}\big\}$. The first elements of these  sets are different.
A: It implies $\langle b,a\rangle=\{{\{{b\}},\{{a,b\}}\}}$, so $\langle a,b\rangle\neq\langle b,a\rangle$. 
