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, 201213 yr by Prometheus

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}.

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.

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.

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

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

No!