Power sets

[Prometheus]

Just started to learn a bit of set theory, and was hoping someone could aid my understanding of power sets.

 

As i understand it a power set is the set of all subsets within a given set.

This includes the null set - easy enough. It also includes each of the elements within the set. Isn't this saying that each element of a set is also a subset of the set?

It then includes the set of all elements. Since all the elements are the set, isn't this saying that set A is a subset of itself?

 

Hopefully i'm just confusing an element with a subset, but any help is appreciated.

 

P.S.

 

And am i right in thinking there can be no subsets of the power set, only elements?

Edited October 5, 2012 by Prometheus

[mathematic]

I'll show by example: S = {a,b,c}, Power set = { {}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

 

Notice the power set does not have a,b, or c, but it does have the single element subsets {a}, {b}, and {c}.

[QuantumBullet]

It then includes the set of all elements. Since all the elements are the set, isn't this saying that set A is a subset of itself?

 

 

 

 

The set A is a subset of the Power Set A. It is also an element of the Power Set A.

[mathematic]

It then includes the set of all elements. Since all the elements are the set, isn't this saying that set A is a subset of itself?

 

 

 

 

The set A is a subset of the Power Set A. It is also an element of the Power Set A.

There is a distinction. In the example I gave, S={a,b,c} is an element of the power set of S. {S} = {{a,b,c}} is a subset of the power set.

[QuantumBullet]

In that example, {a,b,c} is both a distinct element, and a subset, of the Power Set.

[mathematic]

In that example, {a,b,c} is both a distinct element, and a subset, of the Power Set.

No!