a$.$ Suppose a sample space is defined by the Cartesian product of the ranges of a set of

Boolean variables $x_1,$dots,$x_n,$ and let $$ be any logical proposition expressed in terms

of these variables. Prove from first principles $($including the basic axioms of probability,

Equation $(12\mathrm{.}1))$ that Equation $(12\mathrm{.}2),$ i$.$e$.,P()={\displaystyle \underset{in}{\overset{?}{}}}P().$

b$.$ Now show that ${\displaystyle \underset{x_i}{\overset{?}{}}}P(x_i=x_i)=1$ for any variable $x_i,$ i$.$e$.,$ the distribution for each

random variable must sum to $1.$