Question about Kuratowski Definition of Ordered Pairs

Question

I just finished high school and started my journey with self-learning Maths. I have a question about Kuratowski Definition of Ordered Pairs: (a,b)={{a},{a,b}} I'm not quite sure what this means exactly even though I looked through multiple questions here and on quora. Is this (the right side) just a notation to indicate which element comes first? or there's actually some set-theory-logic behind it?

Answer 1

Often in Mathematics, concepts arise from an intuition, but the formalisation is another story, often improved constantly through experience and the passage of time.

Without diving into the formalities, we just want a pair of elements with an order defined.

Why would we do that? Because it is interesting in the study of certain problems or areas of mathematics. It's not the same to walk north 3 metres and then 6 metres to the east, in contrast to walking 6 metres to the north and 3 metres to the east. There is notion of order, and we could think of $(3, 6)$.

However, mathematicians have been striving to build a solid ground for the building of mathematics, and in that effort, the set theory was born, the heart of all mathematics. So, it is natural that mathematicians try to define this thing called "$(a, b)$" using only the basics: the sets. So, in the end, this definition is just formality, you don't need it to understand what an ordered pair is.

Jumping back to the subject, notice that the set $\{3, 6\}$ is lacking when we try to represent this information. One would think a way to sort this out by defining a new set $\{\{3, 1\}, \{6, 2\}\}$. And for this case, it works. Eureka! Now we can convey the order using only sets! (This way of defining an ordered pair is called Hausdorff's definition). However, you might notice that it wouldn't work if we had the ordered pair $(2, 1) = \{\{2,1\},\{1, 2\}\} = \{1,2\}$.

Many definitions would fail with the passage of time, until some prevailed over the others, because they proved that they worked. One of such is Kuratowski's definition. How can we define order using ONLY sets and the elements given?

Kuratowski answers this question by defining $(a, b) = \{\{a\}, \{a, b\}\}$ . As you can see, the set $\{a, b\}$ conveys the information on what elements are present on the pair, and the set $\{a\}$ conveys the information on who is the first element.