5
$\begingroup$

I know that you can represent an ordered pair using the ZFC axioms like that : $\{\{x\},\{x,y\}\}$. The question is :

Prove that it is possible to represent an ordered pair in the form of : $\{x,\{y\}\}$.

I need to show that $\{x,\{y\}\} = \{a,\{b\}\}$ if and only if $x = a \wedge y = b$.

The first side of the proof is trivial because all you need to do is to place $x$ instead of $a$ and $y$ instead of $b$.

I'm stuck in the other side of the proof. If dividing it into $2$ cases :

  • If $x = a$ than $\{y\} = \{b\}$ and thus $y = b$.
  • If $x = \{b\}$ than $\{y\} = a$, and this is where I'm stuck at.

Thanks in advance !

$\endgroup$
  • 2
    $\begingroup$ To the person who voted to close this as "missing context or other details": I do not understand this. The question is perfectly clear. We often vote to close as "missing context" when OP pastes a homework question with no indication of their background or what their question is, but that did not happen here. $\endgroup$ – MJD Mar 8 '15 at 15:56
  • $\begingroup$ @MJD someone is rampaging through everything and voting to close with this reason. There are 50 close votes in the queue at a time, probably all from this person. $\endgroup$ – Matt Samuel Mar 8 '15 at 16:29
10
$\begingroup$

It isn't true. Consider $\{\{a\},\{b\}\}$. It can be viewed either as $\langle \{a\},b\rangle$ or as $\langle \{b\},a\rangle$, according to your new definition of ordered pair. Since these can be different, this definition doesn't have the ordered-pair property.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.