Question about set theory and ordered pairs

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The discussion revolves around the set-theoretic definition of ordered pairs, specifically Kuratowski's definition, which uses the formulation (x,y)={{x},{x,y}}. This definition allows for the identification of the first and second elements without relying on their order. The inquiry raises concerns about an alternative definition, (x,y)={x,{y}}, questioning its validity due to potential counterexamples where pairs could be considered equal despite differing orders. The example provided illustrates that under the simpler definition, pairs like ({0},1) and ({1},0) could be mistakenly deemed equivalent. The consensus emphasizes the necessity of the more complex definition to maintain the integrity of ordered pairs in set theory.
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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?
 
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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
 

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