Show that: Let A be a set and let $P(A)$ be the set of all subsets of $A$. Then there is no surjection $f: A→P(A)$.
Here is what I thought:
if $A=\{a,b\}$ then it has only two elements where $P(A)=\{∅,\{a\},\{b\},\{a,b\}\}$ has 4 elements. Therefore $f:A→P(A)$ cannot be surjective. But I have some problems:
1) How is it possible that any $f$ function to take $\{a\}$ from set $A$ to $\{a,b\}$? Maybe because I am thinking mainly about functions with real values like $f(x)=2x$, I find it a little bit strange that a function to take an element of a set to another set which has more elements. Is it possible?
edit: Now I thought that if $f(x)$ is $\sqrt{x}$, then $f(4)=±2$ which means it took an element from a set to a set which has 2 elements. But still I find it kind of strange to denote $f(\{a\})=\{a,b,c,...\}$
2) How can I construct a explicit proof for this question?
Regards