R is a partial order relation on some set A. Which of the following statements are correct?
a) $R\cup R^{-1}$ is an equivalence relation
b) $R^{2}$ is a partial order relation
c) $R\cap R^{-1}$ is an equivalence relation
I've tried solving this. For (a), I think this is true, but the answer I got say false. My logic was:
for every x, $xRx$ since R is reflexive. R is also ant-symmetric, so for every pair $xRy$, we will have $yRx$ because of the union, and so $R\cup R^{-1}$ is symmetric. Now I tried thinking on transitive, I took an example:
$R={(1,1),(2,2),(3,3),(1,2),(2,3),)(1,3)}$
If you do $R\cup R^{-1}$ you get a transitive relation. I can't find an example to break it.
What am I doing wrong ?
For (b) I tried a similar example and go $R^{2}=R$ which shows it's true, but it must have been a private case. How can you explain the fact that (b) is true ? (I am not looking for a formal proof).
For (c), if I use intersection, I get only "reflexive" couples, so this is true. Am I correct on this one at least ?
Thank you.
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