# 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?

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.

• Upvoted, but no reason to replace $=$ with $\subset$ except visual similarity of a symbol.
– z100
Commented Oct 6, 2016 at 22:05

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.

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.

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

• I interpret "a set of all natural numbers" to mean $A \subset \mathbb N$. e.g. {2,7,9,15} is a set of all natural numbers, all four of them. Commented Oct 6, 2016 at 23:06