# Question on elementary set theory notation

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?

• $\{na\}$ is just a single set with a single element, which depends on the "external" constants $a$ and $n$. – Hagen von Eitzen Jul 21 at 11:50
• $n$ is not a constant. $A=\{a,2a,3a,4a,....\}$ – Joe Jul 21 at 11:57
• The set-builder notation you're trying to achieve is \{ na \mid n \in \mathbb{N} \}, producing $\{na \mid n \in \mathbb{N}\}$. – Hayden Jul 21 at 12:12

• 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!
• Well, $n$ is not fixed.Your second option is correct. Anyway how can we write the information about $a$, i.e. $a>0$ in the set notation? – Joe Jul 21 at 12:00
• You already fixed $a>0$, so it is also better to write the set by $$A_a=\{na: n \in \Bbb N\}$$ which means the element $a$ is fixed is understood – Chinnapparaj R Jul 21 at 12:06
• or just write $$A=\{na: n=1,2,3, \cdots\;\text{and}\;a>0\;\text{is fixed} \}$$ – Chinnapparaj R Jul 21 at 12:09
You can also write it as $$a\Bbb N$$.