Let A infinite set
$|A|=a$
$B=\{x|x \in P(A)\}$
$C=\{f|f:A \to P(A)\}$
$F=\{$ Set of all the relations on $A \times B \}$
$H=\{g|g:B \to B \}$
1) Find $|C|$
2)Find $|F\times H|$
for (1) i think $|P(A)^A|= |P(A)|^{|A|}=2^a$ is that correct and how to find (2)
thanks
Obviously $|C|=|F|$.
The number of function from one element of $B$ to all the elements of $B$ is $2^{a}$. It means that $|H|=(2^{a})^{2^{a}}$.
These are the solutions: $$|C|=2^{a}*a$$ $$|H \times F|= (2^{a})^{2^{a}}*|F|= (2^{a})^{2^{a}+1}*a$$
These solutions consider all types of function.