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Arbitrary Union of Sets Question

  1. Mar 12, 2016 #1
    1. The problem statement, all variables and given/known data

    For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##.

    2. Relevant equations

    3. The attempt at a solution
    I know that this involves natural numbers some how, I am just confused on a notation thing.
    Is ##\bigcup_{n\in\mathbb{N}}A_{n}= \left \{ 1,2,3,... \right \}## or##\bigcup_{n\in\mathbb{N}}A_{n}= \left \{ \left\{1\right\},\left\{2\right\},... \right \}##.

    To me it makes more sense that this is set of each set of every natural numbers, but somehow I think this is wrong and can't wrap my head around it.
     
    Last edited: Mar 12, 2016
  2. jcsd
  3. Mar 12, 2016 #2
    The problem statement is incomplete, it doesn't describe ##A_n##.

    In terms of notation,
    • ##x \in \cup_{n\in\mathbb{N}} A_n ## if and only if there exists ##n\in\mathbb{N}## such that ##x\in A_n## (##x## belongs to at least one of the ##A_n##).
    • ##x \in \cap_{n\in\mathbb{N}} A_n ## if and only if ##x\in A_n## for all ##n\in\mathbb{N}## (##x## belongs to all ##A_n##).
     
  4. Mar 12, 2016 #3
    I updated it to define An.
     
  5. Mar 12, 2016 #4

    Mark44

    Staff: Mentor

    ##\bigcup_{n\in\mathbb{N}}A_{n}## means ##A_1 \cup A_2 \cup \dots \cup A_n## and similar for the intersection.
     
  6. Mar 12, 2016 #5
    Ok so you infered that the union was equal to ##\mathbb{N}##. How do you prove that two sets are equal ?
     
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