$\
ewcommand{T}{{\rm T}} \
ewcommand{S}{{\rm S}} \
ewcommand{K}{{\rm K}} \
ewcommand{U}{{\rm U}} \
ewcommand{and}{\text{ and }}$
> All Texans speak to anyone whom they know intimately.
Rephrase it as "If a x is a Texan, and if x knows y, then x speaks to y".
> $\forall x, y \quad \T x \and \K xy \Longrightarrow \S xy$
* * *
No Texan speaks to anyone who is not a Southerner.
Rephrase it as "If x is a Texan and if y is not a southerner, then x does not speak to y."
> $\forall x, y \quad \T x \and \lnot \U y \Longrightarrow \lnot S xy$
or if you wish to be advanced:
> $\bigg\vert\\{x ~\vert~ \T x \and \exists y ~ \lnot \U y \and \S x y\\}\bigg\vert = 0$
* * *
> Therefore, Texans know only southerners intimately.
Rephrase it as "If x is Texan, and x knows y, then y is a southerner."
> $\forall x, y \quad \T x \and \K xy \Longrightarrow U y$
* * *
To prove the 3rd rule using the first 2, transform the 2nd rule to it's contrapositive.