(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Define a relation on the plane by setting:

[tex] (x_o,y_o)<(x_1,y_1) \mbox{ if either } y_0-x_0^2<y_1-x_1^2, \mbox{ or } y_o-x_o^2=y_1-x_1^2 \mbox{ and } x_0<x_1.[/tex]

I have easily showed that Nonreflexivity and Transitivity hold. The only dilemma i am facing is to show that Comparability holds as well.

That is to show that for any two elements

[tex](x_0,y_0),(x_1,y_1)\in R^2[/tex]

such that :

[tex](x_0,y_0)\not=(x_1,y_1)[/tex]

then either

[tex] (x_0,y_0)<(x_1,y_1) \mbox{ or } (x_1,y_1)<(x_0,y_0)[/tex]

To be more specific, is proof by cases the only way to go about it, or is there any way around it? There seem to be too many cases, and i don't really want to pursue this route. I am also thinking about proving its contrapositive, but still, it looks like i would have to treat a few cases separately.

Any suggestions would be appreciated!

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# Homework Help: Linear Order on RxR. ?

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