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.
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
$\displaystyle R^U$ is the converse, or the inverse, of $\displaystyle R$, means that if the element $\displaystyle (x, y)$ in R, the element $\displaystyle (y,x)$ is in $\displaystyle R^U$ - 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 $\displaystyle R^{-1}$, as you might expect).
So, if $\displaystyle R$ is reflexive and transitive, all you'll need to prove in order for $\displaystyle R\cup R^U$ to be an equivalence relation is:
- $\displaystyle R^U$ is also reflexive and transitive, and therefore $\displaystyle R \cup R^U$ is reflexive and transitive
- $\displaystyle R \cup R^U$ is symmetric; in other words, $\displaystyle (x, y) \in R \cup R^U \Rightarrow (y,x) \in R \cup R^U$
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 $\displaystyle R^U$ is reflexive.
Assume that $\displaystyle R$ and $\displaystyle R^U$ are defined on the set $\displaystyle S$.
Then $\displaystyle R^U$ is reflexive $\displaystyle \iff (x, x) \in R^U, \forall x \in S$.
Now $\displaystyle (x, y) \in R^U \Rightarrow (y, x) \in R$. Therefore $\displaystyle (x, x) \in R^U, \forall x\in S$, since $\displaystyle (x,x) \in R$, because $\displaystyle R$ is reflexive.
$\displaystyle \Rightarrow R^U$ is reflexive.
Next, we prove that $\displaystyle R^U$ is transitive.
$\displaystyle R^U$ is transitive $\displaystyle \iff ((x,y) \in R^U \wedge (y,z) \in R^U) \Rightarrow (x,z) \in R^U$
Now $\displaystyle (x,y) \in R^U \Rightarrow (y,x) \in R$, and $\displaystyle (y,z) \in R^U \Rightarrow (z,y) \in R$
$\displaystyle \Rightarrow (z, x) \in R$, since $\displaystyle R$ is transitive
$\displaystyle \Rightarrow (x, z) \in R^U$
$\displaystyle \Rightarrow R^U$ is transitive
Finally, we prove that $\displaystyle R \cup R^U$ is symmetric.
$\displaystyle (x,y) \in R \cup R^U \Rightarrow (x,y) \in R$ or $\displaystyle (x,y) \in R^U$
Now if $\displaystyle (x,y) \in R$, then $\displaystyle (y,x) \in R^U$, and if $\displaystyle (x,y) \in R^U$, then $\displaystyle (y,x) \in R$.
Either way, $\displaystyle (x,y) \in R\cup R^U \Rightarrow (y,x) \in R\cup R^U$
So $\displaystyle R\cup R^U$ is symmetric.
Finally, then, $\displaystyle R$ and $\displaystyle R^U$ are both reflexive and transitive, and $\displaystyle R\cup R^U$ is symmetric. So $\displaystyle R\cup R^U$ 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