# A functions proof

• Aug 10th 2010, 12:35 AM
nzmathman
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.
• Aug 10th 2010, 12:39 AM
Swlabr
Quote:

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 $|A| < |\mathcal{P}(A)|$?
• Aug 10th 2010, 01:20 AM
nzmathman
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!
• Aug 12th 2010, 02:10 AM
aman_cc
You may want to refer to this thread - I had the same question sometime back
http://www.mathhelpforum.com/math-he...et-113875.html