= Solution
Suppose $F:X\to\mathcal P(X)$ were a <surjection> onto the <power set> of the <set> $X$. Form the <subset>
$$
D=\{x\in X:x\notin F(x)\}.
$$
Surjectivity would give $a\in X$ with $F(a)=D$. But then $a\in D$ holds exactly when $a\notin F(a)=D$, a <contradiction>. Hence no such <surjection> exists, and in particular \b[there is no bijection between a set and its power set]. This is the diagonal construction in <Cantor's theorem>; it also covers the <empty set>, whose <power set> has one element.
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