Help with expressing statements using quantifiers I was hoping for someone to check over my answers for a question I am not totally confident about. I'm also not sure if they would require nested quantifiers (which I didn't use). Thank you in advance.
Let $P(x), Q(x), R(x)$, and $S(x)$ be the statements:


*

*$P(x): x$ is a duck

*$Q(x): x$ is one of my poultry

*$R(x): x$ is an officer

*$S(x): x$ is willing to waltz


The Domain is the set of all poultry. Express each of the statements using quantifiers; logical connectives; and $P(x), Q(x), R(x),$ and $S(x).$


*

*No ducks are willing to waltz.
∀x,(P(x) → ¬S(x))


*No officers ever decline to waltz.
∀x,(R(x) → S(x))


*All of my poultry are ducks
∀x,(Q(x) → P(x))


*My poultry are not officers
∀x,(Q(x) → ¬R(x))


*Is it possible that I have poultry that is willing to waltz? Explain
briefly.
∃x,(Q(x) → S(x)) ...Not quite sure how to explain


Your answers for 1–4 look correct to me.  There's no need for nested quantifiers here, since all the relations are unary (have one variable).
I think that question 5 is asking you to decide if the statement is true assuming that statements 1–4 are true.
