Proof of lemma about power set and one question.

Question

I am new here. I am during my PhD studies in theoretical physics and in order to understand one property of given theory, I need to understand proof of lemma:

$|P(A)|=|\{0,1\}^A|$.

To understand topological aspects of filed theory, that I am investigating, I require more knowledge about set theory. Could you recommend me some books ? My passion is pure mathematical physics, so I am not afraid of advanced calculations, but I am type of person, who needs every proof written clearly.

I have earned a Master's degree in technical physics ( although my thesis was about differential geometry in quantum systems) and sometimes I lack pure mathematical knowledge and have to fill the gap on my own.

Thank you in advance.

Answer 1

If $f$ in $\{0,1\}^A,$ then $A_f = \{ x | f(x) = 0 \}$ is a subset of $A$.
Show the correspondence $f \to A_f$ is the desired bijection.
Such functions are called charactistic functions of the subsets of A.

Answer 2

If $P(A)$ is the set of all subsets of $A$, then you can define a function $I_S$ with domain $A$ for each $S \in P(A)$, such that $I_S(x) = 1\{ x \in S \}$, where $1\{\cdot\}$ is the indicator function which has value in $\{0, 1\}$, such that it equals 1 when the predicate inside has true value, and $0$ otherwise. There are as many subsets as such functions, so the cardinality of both sets is the same.

Now, how many such functions are there ? They take value in $A$ and output in $\{0, 1\}$, so that there are $2^{|A|}$ of them.

Answer 3

There is a function $\phi:\{0,1\}^A\to\wp(A)$ prescribed by $f\mapsto\{a\in A\mid f(a)=1\}$.

This function can be shown to be invertible (so bijective).

Prescribe $\psi:\wp(A)\to\{0,1\}^A$ by stating that $\psi(B)$ is the function $A\mapsto\{0,1\}$ prescribed by $a\mapsto1$ if $a\in B$ and $a\mapsto0$ otherwise.

Then evidently: $$\psi\circ\phi=\mathsf{id}_{\{0,1\}^A}\text{ and }\phi\circ\psi=\mathsf{id}_{\wp(A)}$$