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?

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

Your counterexample looks valid to me.

As for what goes wrong, you may have to ask whoever is claiming (falsely) that the definition works what their response to the counterexample is.

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.


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