anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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Zarkon
  • Zarkon
I'm not following what you problem is with this definition
anonymous
  • anonymous
It says, it is the set of all subsets of A. However, I don't see where 'all' comes from.
anonymous
  • anonymous
ie. all possibilities of subsets of a set

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Zarkon
  • Zarkon
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
anonymous
  • anonymous
Ok so it's implicitly stated because of P?
Zarkon
  • Zarkon
sure P(A) = { X: X ⊆ A} and the set of all subsets of A. are two ways to say the exact same thing.
anonymous
  • anonymous
What does B = {X : X ⊆ A} mean?
Zarkon
  • Zarkon
it says that B is the powerset :)
Zarkon
  • Zarkon
of A
anonymous
  • anonymous
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?
Zarkon
  • Zarkon
no
Zarkon
  • Zarkon
if A={1,2,3} and B=P(A) then \[B=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}\]
anonymous
  • anonymous
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!
Zarkon
  • Zarkon
if \[B=\{X|X\subseteq A\}\] then B is the same as P(A)
Zarkon
  • Zarkon
also you should stop writing B ={{1},{1},{1}} because it make sense. you don't have repeated elements in a set.
Zarkon
  • Zarkon
*because it make no sense
anonymous
  • anonymous
ah yeah of course. Thanks for clearing things up. :)
anonymous
  • anonymous
I'll be sure to ask more questions later

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