- #1
sutupidmath
- 1,630
- 4
Homework Statement
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!