Don't feel ignorant. Just a difference in notation.

Definitions:

- power set of

. In other words it is the set of all functions that map from

__into__ .

- This is just the charceristic function of

. It equals 1 if

and

if not.

equipotent- Two sets

are equipotent (denoted

) if there exists a bijection between the two, i.e.

.

Therefore what one means when they say "

given by

" what they are really saying is "define a mapping between the power set of

and the set of all functions that map

into

by a subset of

(lets say E) is mapped to the charceristic function of that subset (

). Doing this we can easily prove that

. Therefore instead of proving that

and

aren't equipotent you merely need to show that

and

aren't equipotent (this is because equipotence is an equivalance relation). This is actually an easier task.

Here is how. I will give you the outline and you fill in the "why?" in the last step.

Let

be some injective mapping from

to

. Then

, and since

we see that

. Now define a second mapping

by

. Then clearly

but

(why?). Therefore we may conclude that there is no surjection between

and

. Consequently, there is no surjection between

and

. Therefore

*Remark:* Realizing that

given by

is an injection, we may conclude that

. Combining this with teh above we may definitively say that

I hope that helps.