Isomorphism from ring to ring. Exercise from General Topology (J. Kelley)

I tried Exercise K from chapter 2 of General Topology by Kelley. Is this enough to show isomorphism between Boolean rings? This is part (d).

Let $X$ be a set and let $Bbb Z_2^x$ be the family of all functions on $X$ to $Bbb Z_2$ , Define addition and multiplication of functions pointwise:

• $(f + g)(x) = f(x) + g(x)$, and
• $(f cdot g) (x) = f(x) cdot g(x)$.
  Then $(Bbb Z_2^x ,+,cdot)$ is a Boolean ring with unit and is isomorphic to $(mathcal a =mathcal P(X),Delta, bigcap)$.

Assume $(mathcal a =mathcal P(X),Delta, bigcap)$ is Boolean ring (followed from previous exercise).

Proof. Observe that any function of $Bbb Z_2^x$ has value of either $0$ or $1$.
So let $f,gin Bbb Z_2^x$, then for any $xin X$ sum of the functions and multiplication is in ${0,1}$ for any $xin X$:

• If $f(x)=0$ then $f(x)+f(x)=0+0=0,$
• if $f(x)=1$ then $f(x)+f(x)=1+1=2 underbrace{=}_{text{(mod 2)}}0$.

So in all cases, the addition satisfies definition of Boolean ring, which is $r+r=0 quad forall rin R$.

Likewise with multiplication in Bool. ring: $rcdot r=r$. If $f(x)=0$, then $0cdot 0=0=f(x)$, and $1cdot 1=1$ for $f(x)=1$.

Now to show isomorphism between $(Bbb Z_2^x ,+,cdot)$ and $(mathcal a,Delta, bigcap)$ let

$$varphi: Bbb Z_2^x to mathcal a.$$

Then $forall fin Bbb Z_2^x$:

$begin{align}
bullet varphi(f(x)+f(x))=varphi(0)=emptyset&=ADelta A \
&=varphi(f(x)) Delta varphi(f(x)),\
bullet varphi(f(x)cdot f(x))= varphi(f(x))=A&=Abigcap A\
&=varphi(f(x))bigcap varphi(f(x)) \
&= varphi(f(x)cdot f(x)). square
end{align}$