The set of all sets contradiction

[Unassuming]

Homework Statement

Let A be the set of all sets.

#1.) Show that P(A) is a subset of A.

#2.) Find a contradiction.

Homework Equations

The Attempt at a Solution

#1.) We know that P(A) is a set. Therefore it must be in A, since A is the set of ALL sets.

#2.) I cannot figure out what to do here. I do not have a theorem that says that |P(A)| > |A|. I do have a theorem that states that there is no injection from P(A) to A and also that there is no surjection from A to P(A).

Since there is no injection from P(A) to A, does this alone mean that |P(A)| > |A| ?

 

[Dick]

Yes, it does. If |C|<=|D| then there is an injection from C->D. If there is no injection then the opposite must hold so |C|>|D|.