Using the Boolean specification and implementation listed below, prove the

following:

Theorem $1.$ Let $x$ be any non$-$empty set. The Enoal$-$algebra $2^x$ of power sets of $x$

satisfies the Boolean algebra axioms; e$.$g$.x|=T.$

To prove this theorem, show that each of the five axioms listed in $$ hold when

interpreted by the $-$algebra $2x.$ The tricky part is demonstrating for Axiom $5$ that

for any subsets $L_V{?}_{?}$ of $x$ the equation $UV=\frac{x}{\frac{?}{?}}(\frac{x}{\frac{?}{?}}U\frac{x}{\frac{?}{?}}V)$ holds. You will need to

show this equation holds for each of the following cases: ainU,$ain{V}_{,}$ainU$V,a$

$\frac{?}{\text{i}\text{n}\text{U}}V.$

Signature $$

Sorts Bool

Constants False : $$ Bool, True : $$ Bool

Operations and :$B{l}^2$ Bool

or : Bool ${?}^2$ Bool

not : Bool $$ Bool

Axioms $T$

Let abin Bool

true