Does this make mathematical sense?

Question

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?

Answer 1

A couple of things, if $A$ is the empty set doesn't exists any $a\in A$. If $A$ is the set of all natural numbers than you have $A= \mathbb N$. The inclusions are ok.

Answer 2

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.

Answer 3

You have written:

$$a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}$$

and told us to assume $a\in A$ and $A=\mathbb{N}$. Under that assumption, the inclusion $A \in \mathbb{N}$ is incorrect; the set of all natural numbers is not a natural number (sorry I don't have a reference handy for this elementary fact). The other inclusions are correct. If you replace $A \in \mathbb{N}$ with $A\subset \mathbb{N}$, then everything becomes correct.

Answer 4

Just correct a typo as OP stated "A is a set of all natural numbers": $$a \in A = \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}$$