1. ## A functions proof

Hi, I have a question I am having trouble starting:

Let A be a nonempty set. Prove there is no function A onto P(A) (power set of A).

I am not sure if it just means a general function f:A -> P(A) or if we are talking about a function that is onto. Any help will be greatly appreciated.

2. Originally Posted by nzmathman
Hi, I have a question I am having trouble starting:

Let A be a nonempty set. Prove there is no function A onto P(A) (power set of A).

I am not sure if it just means a general function f:A -> P(A) or if we are talking about a function that is onto. Any help will be greatly appreciated.
Yes, the question is asking about a surjection. Otherwise, you could take the function to be $x\mapsto \{x\}$.

To prove this result, you can use arguments based on the cardinality of $\mathcal{P}(A)$. Now, do you know about the rather handy relationship between $|\mathcal{P}(A)|$ and $|A|$ which states that $|A| < |\mathcal{P}(A)|$?

3. It was just the wording I was slightly confused by, but your example x->{x} made it very clear what the question wanted me to do, and I think I have proved it o.k. now. Thanks!

4. You may want to refer to this thread - I had the same question sometime back
http://www.mathhelpforum.com/math-he...et-113875.html