Is the property reflexive, symmetric, anti-symmetric, transitive, equivalence relation, partially ordered given the relation below? I'm working on this and I'm supposed to figure out if the following properties apply to the below relations.
Properties are:
1. Reflexive
2. Symmetric
3. Anti-Symmetric
4. Transitive
5. Equivalence Relation
6. Partially Ordered Set

Relation:


*

*The relation $R$ on the set of all real function $f: \mathbb{N} \to \mathbb{R}^+$ where $f \ R \ g$ if and only if  $f(n) = O(g(n))$


*The relation $R$ on the set of all real function $f : \mathbb{N} \to \mathbb{R}^+$ where $f \ R \ g$ if and only if  $f(n) = \Theta(g(n))$

My work so far:
For first relation:

a. YES:
b. NO:
c. NO:
d. YES:
e. NO:
f. NO:

For second relation

a.YES:
b.YES:
c.NO:
d.YES:
e.YES:
f.NO:

Am I doing this right? Thank you so much for any help.

If I were correcting your (presumably) homework, I would want more details on your reasoning for transitivity, in both cases. Nevertheless, all your answers are correct. Good job.
