Is the property reflexive, symmetric, anti-symmetric, transitive, equivalence relation, partially ordered given the relation below?

Question

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:

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

2. 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.

Answer 1

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.