On Kuratowski's Ordered Pairs

Consider a hypothetical scenario in which nobody on Planet Z knows (hence doesn't care!) what an order is. Some alien material was found at an unknown farm. Name this matter '%'.

After a few weeks, it was reported that another kind of alien material has been found on some other distant farm, let name it '#'. In similar fashion, as time passed, we get to know about various alien matters, viz. '&', '*', '/', 'y' and '5'  in nearby regions. These findings can be categorized as follows-

• Week 1 Total findings : '%'

• Week 3 Total findings : '%', '#'

• Week 7 Total findings : '%', '#', '&'

• Week 15 Total findings : '%', '#', '&', '*'

• Week 20 Total findings : '%', '#', '&', '*', '/'

• Week 21 Total findings : '%', '#', '&', '*', '/', 'y'

• Last Week Total findings : '%', '#', '&', '*', '/', 'y', '5'

This data was made public so anybody who wills to read it, would be able to decipher what actually happened in last few weeks. If we create a set which contains the set of total findings of a particular week as the element, then this set will be defined as:

$$\bigcirc = \{ \{\%\}, \{\%, \#\}, \{\%, \#, \&\}, \{\%, \#, \&, *\}, \{\%, \#, \&, *, /\}, \{\%, \#, \&, *, /, y\}, \\ \{\%, \#, \&, *, /, y, 5\}\}$$

so this data is giving them a sort of idea that when happened what. If they would look upon the set, it is noticeable that '%' is the only element which is present in each subset of $\{\%, \#, \&, *, /, y, 5\}$ in $\bigcirc$ so it would have been some matter which was found out first, then #, which was also there in each set except first one so it would have been discovered after % and in the sample way, it could be known that which element discovered after what. 

Now how these findings are beneficial for us on Earth? Unlike Planet z, we have a concept of order, so in our set theoretical language, we would say that $\bigcirc$ is in itself a representation of an ordered set of findings. Another interesting and powerful characteristic of $\bigcirc$ is that you don't even need to know the week a particular total finding belongs to i.e. instead of earlier bullet-points, if you were given the following :

• Week A Total findings :  '%', '#', '&', '*', '/', 'y', '5'

• Week B Total findings : '%', '#', '&'

• Week C Total findings : '%' 

• Week D Total findings : '%', '#'

• Week E Total findings : '%', '#', '&', '*', '/'

• Week F Total findings : '%', '#', '&', '*', '/', 'y'

• Week G Total findings :  '%', '#', '&', '*'

(where A, B, C, D, E, F, and G are not necessarily in the alphabetic order) 

Then your set would have been

$$\bigcirc = \{ \{\%, \#, \&, *, /, y, 5\}, \{\%, \#, \&\}, \{\%\}, \{\%, \#\}, \{\%, \#, \&, *, /\}, \{\%, \#, \&, *, /, y\}, \\ \{\%, \#, \&, *\}\}$$

Surprisingly, you can still tell using the same logic, which finding happened first, which one happened later and which happened in the last and in between.

It means that every order has a corresponding set so that if you don't know what an order is, you can still analyse it's set to get an idea of the information provided by an order without actually knowing the very word itself. Similarly if only two elements $x$ and $y$ are considered then the set corresponding to the ordered pair of these two elements will be :

$$A = \{\{x\}, \{x, y\}\}$$ 

In other words, if $(x, y)$ is an ordered pair then it can be defined in terms of a set as follows:

$$(x, y) = \{\{x\}, \{x, y\}\}$$

This definition was given by polish mathematician and logician Kazimierz Kuratowski and the ordered pairs defined this way are known as Kuratowski's ordered pairs.