Question

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.

Answer 1

I'm not following what you problem is with this definition

Answer 2

It says, it is the set of all subsets of A. However, I don't see where 'all' comes from.

Answer 3

ie. all possibilities of subsets of a set

Answer 4

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

Answer 5

Ok so it's implicitly stated because of P?

Answer 6

sure

P(A) = { X: X ⊆ A}
and
the set of all subsets of A.

are two ways to say the exact same thing.

Answer 7

What does B = {X : X ⊆ A} mean?

Answer 8

it says that B is the powerset :)

Answer 9

of A

Answer 10

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?

Answer 11

no

Answer 12

if A={1,2,3} and B=P(A)

then \[B=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}\]

Answer 13

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!

Answer 14

if \[B=\{X|X\subseteq A\}\]

then B is the same as P(A)

Answer 15

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

Answer 16

*because it make no sense

Answer 17

ah yeah of course. Thanks for clearing things up. :)

Answer 18

I'll be sure to ask more questions later