1
$\begingroup$

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} $ }
$\endgroup$
1
  • 1
    $\begingroup$ $\{x \in \emptyset : [\text{anything here}] \}$ 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
    Commented Apr 12, 2017 at 4:27

2 Answers 2

1
$\begingroup$

$\Bigl\{ \{n\cdot k \mid k\in\mathbb{N}\} ~~~~ {\large\mid} ~~~~n\in\mathbb{N} \Bigr\}$

$\endgroup$
0
$\begingroup$

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}

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .