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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} $ }
2 Answers
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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}
