I am having hard time understanding the proof that there is no onto function from set A to its power set. In the proof author has constructed the set B = {$a \in A : a \notin f(a)$}. I have taken few sets and constructed functions but i am not able to get the intuition behind construction of set B , i mean to say how should we know that set B has to be constructed in this way
Thanks
It is the beautiful idea of the proof to look at exactly this set $B$. Don't mind when you think you might not have had this idea.
We have to show that no function $f:\>A\to{\cal P}(A)$ can be surjective. Cantor's idea is to produce for any such $f:\>A\to{\cal P}(A)$ a set $B_f\in {\cal P}(A)$ that does not occur as function value $f(a)$ for some $a\in A$. The set $B_f$ he proposes is $$B_f:=\bigl\{a\in A\bigm| a\notin f(a)\bigr\}\ .$$ It is then an exercise for the reader to prove that $B_f\ne f(a)$ for all $a\in A$.