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anonymous

  • 5 years ago

My book on Sets says: If A is a set, then P(A) = { X: X ⊆ A} is called the power set. It is the set of all subsets of A. However the definition seems to say, every in P(A) is a subset of A, but they may be the same subset. Basically, the second sentence of the definition does not ring true to me.

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  1. Zarkon
    • 5 years ago
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    I'm not following what you problem is with this definition

  2. anonymous
    • 5 years ago
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    It says, it is the set of all subsets of A. However, I don't see where 'all' comes from.

  3. anonymous
    • 5 years ago
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    ie. all possibilities of subsets of a set

  4. Zarkon
    • 5 years ago
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    yes..that is what is is...the set of all subsets of a given set.. P(A) = { X: X ⊆ A} is read as the set of all sets X such that X is a subset of A

  5. anonymous
    • 5 years ago
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    Ok so it's implicitly stated because of P?

  6. Zarkon
    • 5 years ago
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    sure P(A) = { X: X ⊆ A} and the set of all subsets of A. are two ways to say the exact same thing.

  7. anonymous
    • 5 years ago
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    What does B = {X : X ⊆ A} mean?

  8. Zarkon
    • 5 years ago
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    it says that B is the powerset :)

  9. Zarkon
    • 5 years ago
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    of A

  10. anonymous
    • 5 years ago
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    But couldn't this mean say A is {1,2,3} that B = {{1},{1},{1}} since it's true that all elements are subsets?

  11. Zarkon
    • 5 years ago
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    no

  12. Zarkon
    • 5 years ago
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    if A={1,2,3} and B=P(A) then \[B=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}\]

  13. anonymous
    • 5 years ago
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    but the notation is different. B = {X : X ⊆ A} is not B=P(A).How would one define the rules for the previous example? Like A = {1,2,3} then B ={{1},{1},{1}} or B could be {{1},{2},{2}} etc as long as the elements were subsets. Thanks!

  14. Zarkon
    • 5 years ago
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    if \[B=\{X|X\subseteq A\}\] then B is the same as P(A)

  15. Zarkon
    • 5 years ago
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    also you should stop writing B ={{1},{1},{1}} because it make sense. you don't have repeated elements in a set.

  16. Zarkon
    • 5 years ago
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    *because it make no sense

  17. anonymous
    • 5 years ago
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    ah yeah of course. Thanks for clearing things up. :)

  18. anonymous
    • 5 years ago
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    I'll be sure to ask more questions later

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