Does this make mathematical sense? For a given set $A$, 
An element such that $a \in A $ exists. 
If $A$ is a set of all natural numbers, then:
$$ a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}. $$
Would maths normally be written like this, if it is correct? 

This question is a bit confusing and no it doesn't make a lot of "sense" overall. Especially given that $A$ being defined as the set of all natural numbers means $A\not\in\mathbb{N}$ but that $A=\mathbb{N}$.
As for your question, would maths normally be written like this... yes, those are all valid mathematical symbols and there is some logic to the way you are formulating them.
