Big Oh and Big Theta relations confirmation I just want to confirm these statements, 
I know that Big O, and Big theta, are partial order and equivalence relations respectively, all positive integers, but not sure on these restrictions.
$f:N \rightarrow R^+$
where $f$ R $g$ is and only if $f(n) = O(g(n))$
This is still a PO (Partial Order)
where $f$ R $g$ is and only if $f(n) = Θ(g(n))$ 
This is still a ER (Equivalence Relation)
Thank you!

You are correct that Big-Oh is a partial order, and Big-Theta is an equivalence relation. One can say that $f < g$ if $f \in O(g)$ (or $f = O(g)$, alternate notation-wise).
Notice though that $\sin(n) \in O(n)$ and $\cos(n) \in O(n)$, so that with this order we have $\sin(n) < n$ and $\cos(n) < n$, but we have neither $\sin(n) < \cos n$ nor $\cos(n) < \sin(n)$. This is why it's a partial order (as opposed to a total order).
The equivalence relation $f \sim g$ iff $f \in \Theta(g)$ is, as you've called it, an equivalence relation.
