Question on elementary set theory notation

Question

Consider the set $A=\{n\ a \}$ where $a>0$ is a constant and $n \in \mathbb{N}$

How shall we write this set $A$ in set theory?

If we write it as $A=\{n\ a\ \backslash n \in \mathbb{N}, a>0 \}$ or $A=\{n\ a\ / n \in \mathbb{N}, a>0 \}$ will it mean just one set or a set of infinite sets?

Answer 1

• If both $n$ and $a$ are fixed, then the set is , the singleton $\{na\}$
• If $a$ is fixed and $n \in \Bbb N$ is a varying quantity, then the set is $\{na: n \in \Bbb N\}=\{a,2a,3a,\cdots\}$

In both cases, $A$ is one set with cardinality $1$ and infinite(of course, $\aleph_0)$ respectively!

Answer 2

You can also write it as $a\Bbb N$.