Set relationship between Big-Oh and Theta notations I was reading &quot;Introduction to Algorithms&quot; by CLRS and it says that:

We write $f(n) = O(g(n))$ to indicate that a function $f(n)$ is a member of of the set $O(g(n))$. Note that $f(n) = \Theta(g(n))$ implies $f(n)=O(g(n))$, since $\Theta$-notation is a stronger notion than $O$ notation. Written set-theoretically, we have $\Theta(g(n)) \subseteq O(g(n))$.

Q1: What do the authors mean by strong notion? What is strong notion, when we use it, how does it help us (here) knowing one implication is stronger than the other and how does it affect when creating implication.

Q2: It seems contradictory in a sense by saying that $\Theta$ is stronger notion than $O$ and then writing $\Theta(g(n)) \subseteq O(g(n))$.How to deduce and then interpret the second sentence from the first?

I would appreciate your answers.
In general, a proposition $P$ is said to be stronger than $Q$ if $P$ implies $Q$ ( symbolically, $P\Rightarrow Q$)  and $NOT$( $Q$ implies $P$).  In the example at hand, $f \in \Theta(g) \Rightarrow f \in \mathcal{O}(g)$ , so that $f \in \Theta(g)$ is a stronger assertion than $f \in \mathcal{O}(g)$.