# When defining ordered pairs, are there any important distinctions between $\{\{a\},\{a,b\}\}$ and $\{a,\{b\}\}$?

The formal Kuratowski definition of ordered pair is that $$\langle a,b\rangle = \{\{a\},\{a,b\}\}$$.

While I think I understand the above definition well I wanted to check if below definition also works just fine (and hence is "equivalent" to Kuratowski definition)

$$\langle a,b\rangle = \{a,\{b\}\}.$$

I think that both the definitions are just fine, but maybe I'm missing a subtle point. Also is there any reason to prefer Kuratowski's definition over the later one?

Unfortunately, with the new definition, both $$\langle\{0\},1\rangle$$ and $$\langle\{1\},0\rangle$$ equal $$\{\{0\},\{1\}\}$$. Thus this definition is not suitable for ordered pairs.
With Kuratowski's definition, $$\langle a,b\rangle=\langle c,d\rangle$$ if and only if $$a=c$$ and $$b=d$$, as we'd hope. However, in the proposed approach $$\langle a,b\rangle=\{a,\{b\}\}$$, observe that $$\langle \{1\},2\rangle = \langle \{2\},1\rangle$$, so we don't have uniqueness.
The ordered pair $$(a,b)$$ is defined in such a way that it satisfies the following:
$$(a,b)=(c,d)\Longleftrightarrow a=c$$ and $$b=d$$.
So once your definition of $$(a,b)$$ satisfies above, it is absolutely fine. But your definition does not satisfy above, so it is not valid. Otherwise you can have any of the following as your definition:
$$(a,b)=\{a,\{a,b\}\}$$ or, $$\{\{b\},\{a,b\}\}$$ or, $$\{b,\{a,b\}\}$$