Constructing Logical Derivation

Question

All Texans speak to anyone whom they know intimately. No Texan speaks to anyone who is not a Southerner. Therefore, Texans know only southerners intimately. (We have to use These predicates : $Tx, Sxy,Kxy,Ux$).
$Tx : x$ is a Texan.
$Sxy : x$ speaks with $y$.
$Kxy :x$ knows $y$.
$Ux: x$ is a southerner.

I can't symbolise these arguements, so question of constructing derivation is far. Please, just explain me in detalis how you would symbolise these arguements.

Answer 1

$\newcommand{T}{{\rm T}} \newcommand{S}{{\rm S}} \newcommand{K}{{\rm K}} \newcommand{U}{{\rm U}} \newcommand{and}{\text{ and }}$

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".

$\forall x, y \quad \T x \and \K xy \Longrightarrow \S xy$

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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:

$\bigg\vert\{x ~\vert~ \T x \and \exists y ~ \lnot \U y \and \S x y\}\bigg\vert = 0$

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Therefore, Texans know only southerners intimately.

Rephrase it as "If x is Texan, and x knows y, then y is a southerner."

$\forall x, y \quad \T x \and \K xy \Longrightarrow U y$

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To prove the 3rd rule using the first 2, transform the 2nd rule to it's contrapositive.