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I understand the fundamentals of set-builder notation, however, a more sophisticated problem with nested sets is stumping me. I'd really like some constructive criticism on how I approach this problem and even some wisdom for future problems. As an example, I'll be using set-builder notation on:

A set with infinitely many elements with each element being a set that itself 
has infinitely elements such that:

{{1,2,3,...}, {2,4,6,...}, {3,6,9,...}, ...}.

My thought process:

  • I need two variables to define two sets to work with
  • One variable, set 1, needs to be the empty set
  • The other variable, set 2, needs to be the set of natural numbers
  • The resulting set is set 2 added to set 1 infinitely many times

My notation:

  • {x $\in\emptyset$: x = {x + y} where y $ \in\mathbb{N} $ }
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    $\begingroup$ $\{x \in \emptyset : [\text{anything here}] \}$ is always $\emptyset$. Can you write each of the "inner" sets using set builder notation? Can you write a "generic" formula for an inner set using set builder notation (it will probably involve a parameter, say $k$)? If you can, let's call the inner set $S_k$ for a moment. Then your set is $\{S_k : k \in \Bbb N\}$, and just replace $S_k$ with its set-builder notation. $\endgroup$ – pjs36 Apr 12 '17 at 4:27
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$\Bigl\{ \{n\cdot k \mid k\in\mathbb{N}\} ~~~~ {\large\mid} ~~~~n\in\mathbb{N} \Bigr\}$

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The given set can be written as

{{x:x is a multiple of 1}, {x:x is a multiple of 2}....}

Therefore the set builder is

{X: X is a set containing the multiples of a given natural number}

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