There are a few definitions of ordered pairs $(a,b)$.
Wiener's def: $(a,b)_{W} = \Big\{ \big\{ \{a\}, \emptyset \} \big\}, \big\{ \{b\} \big\} \Big\}$
Kuratowski's def: $(a,b)_{K} = \big\{ \{a\}, \{a,b\}\big\}$
But neither of these definitions has the associativity property:
$\big(a,(b,c)\big)=\big((a,b),c\big)$.
Is it possible to define $(a,b)$ such that it still has the defining property
$(a,b)=(c,d)$ if and only if $a=c$ and $b=d$
as well as the associativity property?
I was just wondering about this question. My guess is: the answer is no. But I don't know a method of showing that it is not possible to give such a definition of $(a,b)$. If you know a reference, please let me know.
If such definition were posible you'll have the following:
By associativity: $$((a,b),c)=(a,(b,c))$$ but then, by the defining property of the ordered pairs $$(a,b)=a$$ which is absurd since the LHS depends on $b$ and the RHS doesn't.