Answer
a) $\forall x(P(x)\rightarrow \neg S(x))$
b) $\forall x(R(x) \rightarrow S(x))$
c) $\forall x(Q(x) \rightarrow P(x))$
d) $\forall x (Q(x) \rightarrow \neg R(x))$
e) Yes
Work Step by Step
a) Rewrite the given statement " No ducks are willing to waltz." in If-then form.
If x is a duck, then x is not willing to waltz. This statement if true for all x.
$\forall x(P(x)\rightarrow \neg S(x))$
b) Rewrite the given statement " No officers ever decline to waltz in If-then form.
If x is a officer, then x is willing to waltz.This statement if true for all x.
$\forall x(R(x) \rightarrow S(x))$
c)Rewrite the given statement " All my poultry are ducks" in If-then form.
If x is one of my poultry , then x is a duck.This statement if true for all x.
$\forall x(Q(x) \rightarrow P(x))$
d)Rewrite the given statement " My poultry are not officers" in If-then form.
If x is one of my poultry , then x is not an officer..This statement if true for all x.
$\forall x (Q(x) \rightarrow \neg R(x))$
e) Suppose x is one of my poultry.
then from c) x is a duck
then from a) x is not willing to waltz
Assuming x is an officer, then from b) x is willing to waltz.
But x is not wiling to waltz, so x is not an officer.
Thus, we are able to conclude d) from a), b) and c).
