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.
Hello thehollow89
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?
Grandad
I mean I understand the question and what I need to prove, I'm just not sure how I go about proving it. The prof isn't really all that clear. He'll just teach us about the kinds of relations and then ask us to prove something and most kids end up stumped on what method of proof writing he wants...
First we prove that is reflexive.
Assume that and are defined on the set .
Then is reflexive .
Now . Therefore , since , because is reflexive.
is reflexive.
Next, we prove that is transitive.
is transitive
Now , and
, since is transitive
is transitive
Finally, we prove that is symmetric.
or
Now if , then , and if , then .
Either way,
So is symmetric.
Finally, then, and are both reflexive and transitive, and is symmetric. So is an equivalence relation.
Grandad
The is a counter example to the statement in this problem.
Please see: http://www.mathhelpforum.com/math-he...roof-help.html