Question about set theory and ordered pairs

[reb659]

Hi

I was reading through a textbook and I came across the set theoretic definition of an ordered pair (Kuratowski), where (x,y)={{x},{x,y}}, which apparently can be shortened to {x,{x,y}}. This seems to be the standard definition for an ordered pair in set theory so that we can determine both the first and second element using only set theory and no notions of "first" or "second". However, I am wondering why the less complicated definition (x,y)={x,{y}} does not also work.

Can anyone enlighten me?

 

[Office_Shredder]

If {x,{y}}={a,{b}} then either x=a and {y}={b} which is what we want, or
x={b} and a={y}. So if we just pick y and b arbitrarily we can come up with counterexamples to what an ordered pair should satisfy. So for example ({0},1)=({1},0) under your proposed definition