If R is reflexive and transitive, then R U R^U is an equivalence. I need to prove this using algebraic-relation calculations or disprove by counter example. I'm really not understanding this stuff.
is the converse, or the inverse, of , means that if the element in R, the element is in - because that's what the inverse relation is: the relation you get by switching the order of the elements in the relation. (It's usually denoted by , as you might expect).
So, if is reflexive and transitive, all you'll need to prove in order for to be an equivalence relation is:
- is also reflexive and transitive, and therefore is reflexive and transitive
- is symmetric; in other words,
Can you do that, or do you need further help?
Assume that and are defined on the set .
Then is reflexive .
Now . Therefore , since , because is reflexive.
Next, we prove that is transitive.
Now , and
, since is transitive
Finally, we prove that is symmetric.
Now if , then , and if , then .
So is symmetric.
Finally, then, and are both reflexive and transitive, and is symmetric. So is an equivalence relation.
The is a counter example to the statement in this problem.
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