If there is a nonempty set $A$ and the set of its subsets $P(A)$. How can I show that there is no surjective function $f: A \longrightarrow P(A)$? Tip: Observe the elements $a$ within $A$, for which $a$ isn't an element of $f(a)$ As far as I know it is pretty obvious that there is no such function, because there are much more objects in the image set than there are in the domain. So there is no function which has this characteristic, but that is not a formal proof. I hope this is not regarded as a stupid question. Thank you very much for your help!
I don't follow understand what this theorem is saying: "Every function $f: A \to{\cal P}(A)$ is not surjective."   Is this saying so for example let A = {1,2,3} then the power set of A is {},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3} Is this saying that there is no surjective function that takes every element in each of its power sets to atleast 1 element in A?  Is my understanding correct?
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