Cantors Theorem

Can be there an onto function from $A\rightarrow P(A)$?

Assume that there exists an onto function $g:A\rightarrow P(A)$

Construct a set $S= \{x\in A|x\notin g(x)\}$

Since g is onto, an $s\in P(A)$ has a preimage, i.e. $\exists x\in A(g(x)=S)$

Contradiction #

Let’s analyze if $x\in S$

Assume $x\in S$ $x\notin g(x)\Rightarrow x\notin S$ Contradiction!

Assume $x\notin S$ $x\in g(x)\Rightarrow x\in S$ Contradiction!

No onto function exists from $A\rightarrow P(A)$

Therefore, $|A|<|P(A)|$