Is this statement correct $f(n) = \theta(n) \land g(n) = \Omega(n) \Longrightarrow f(n)g(n) = \Omega(n^2)$?

Question

I am having some difficulties understanding what does it mean to "and" $\theta(n)$ and a function $g(n)$, what does it mean in mathematical terms?

Specifically, in the following example, I have to say if the statement is true or false, if false give an example:

$$f(n) = \theta(n) \land g(n) = \Omega(n) \Longrightarrow f(n)g(n) = \Omega(n^2)$$

I think that the final equals depends on the meaning of $\land$ing two functions (or a set and a function).

So, my main doubt is, what does this means: $\theta(n) \land g(n)$?

Answer 1

It just means that if $f(n) = \Theta(n)$ and $g(n) = \Omega(n)$, then $f(n) g(n) = \Omega(n^2)$.

To prove this, note the first one means there exist $a,b \in \mathbb{R}$ such that for $n \ge N_f$ we have $an \le f(n) \le bn$ and the second one means $\exists c \in \mathbb{R}$ such that $g(n) \ge cn$ for all $n \ge N_g$.

Let $N = \max(N_f, N_g)$. Can you prove that $f(n) g(n) \ge d n^2$?