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?

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

  • $\begingroup$ 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? $\endgroup$ – Joe Jul 21 at 12:00
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    $\begingroup$ 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 $\endgroup$ – Chinnapparaj R Jul 21 at 12:06
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    $\begingroup$ or just write $$A=\{na: n=1,2,3, \cdots\;\text{and}\;a>0\;\text{is fixed} \}$$ $\endgroup$ – Chinnapparaj R Jul 21 at 12:09

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

  • $\begingroup$ This especially comes in handy when you have more sets of this kind and do some operations on them! $\endgroup$ – B.Swan Jul 21 at 12:10

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