# Kuratowski definition of ordered pairs and threetuples.

I am trying to show that $\langle x,y,z\rangle := \{\langle x,y\rangle, \langle y,z\rangle\}$ where $\langle x,y\rangle=\{\{x\},\{x,y\}\}$ is a good definition of an ordered 3-tuple, however I believe the following is a counter-example.

Assume $x=z, x \ne y$, then

$\langle x,y,x\rangle=\{\langle x,y\rangle, \langle y,x\rangle\}=\{\langle y,x\rangle, \langle x,y\rangle\}=\langle y,x,y\rangle$

Since I am told it is a good definition, please can you show me where this goes wrong?

• You clearly proved that it is not a good definition. – drhab Feb 19 '16 at 14:27
• Is it possible that the author meant $\langle x,y,z \rangle := \langle \langle x,y \rangle, \langle y,z \rangle \rangle$ ? – Dan Rust Feb 19 '16 at 14:40

On the other hand, $\langle x,y,z\rangle = \{\langle x,y\rangle, \langle y,z\rangle, \langle x,z\rangle\}$ would work -- but to see this one needs to consider all the possibilities for which of the elements could be equal, and this doesn't generalize to 4-tuples anyway, so it's probably not a good idea.