Is this partially ordered set?

Question

We have set $X$ consisting of positive functions $f: \Bbb{N} \to \Bbb{R}^+$. Are the following relations partial order sets?

1) $f(n)Rg(n) ⇔ f(n) = o\big(g(n)\big)$

2) $f(n)Sg(n) ⇔ f(n) = O\big(g(n)\big)$

3) $f(n)Tg(n) ⇔ f(n) = o\big(g(n)\big) ∨ f(n) = g(n) ∀ n ∈ \Bbb{N}$

I know that partially ordered set means the set is reflexive, transitive and antisymmetrical. But how do I check each one of this qualities?

Answer 1

HINT

Let's do this to $S$, for example.

• reflexive means (f,f) \in S. Is it true that $f(n) = O(g(n))$?
• transitive means that if $(f,g) \in S$ and $(g,h) \in S$ then $(f,h) \in S$. So if $f(n) = O(g(n))$ and $g(n) = O(h(n))$, can you say $f(n) = O(h(n))$?
• antisymmetric means if $(f,g) \in S$ and $(g,f) \in S$ then $f=g$. So if $f(n)=O(g(n))$ and $g(n) = O(f(n))$ then $f(n) = g(n)$?