Logical Predicates

Question

Say you have the following predicates and you're using variables: P(x) = "x is a cow", Q(x) = "x makes milk", and lastly R(x,y) "x and y are the same object".

From there you have the following questions

(a) There are no cows.
(b) Not everything is a cow
(c) There exists at least one cow
(d) There exist at least two cows.
(e) There is at most one cow.
(f) There is exactly one cow.
(g) There is milk producing cow.
(h) Cows produce milk.

Here is what I have:
(a) $\exists x:\neg P(x)$
(b) $\forall x: \neg P(x)$
(c) $ \exists x: P(x)$
(d) $ \exists x \exists y: [P(x) \wedge Q(x) \wedge Q(x) \wedge P(x) \wedge x \ne y] $

e through h im unsure of and I want to know if I made any mistakes so far?

Answer 1

1. "There are no cows" $\equiv$ "It is false that there are cows" $\equiv \, \sim \exists x, P(x)$
2. "Not everything is a cow" $\equiv$ "It is false that everything is a cow" $\equiv \, \sim \forall x, P(x)$
3. "There exists at least one cow" $\equiv$ "There exists a cow" $\equiv \, \exists x, P(x)$
4. "There exists at least two cows" $\equiv$ "There exists a cow $x$ and a cow $y$ such that $x$ is not equal to $y$" $\equiv$ "There exists $x$ and $y$ such that $x$ is a cow, $y$ is a cow, and $x$ is not equal to $y$" $\equiv \, \exists x, \exists y, \, P(x) \wedge P(y) \wedge \sim R(x,y)$.
5. "There is at most one cow" $\equiv$ "Any two cows are the same" $\equiv$ "For all $x$ and $y$, if $x$ and $y$ are cows, then $x$ is equal to $y$" $\equiv \, \forall x, \forall y, \, (P(x) \wedge P(y)) \to R(x,y)$
6. "There is exactly one cow" $\equiv$ "There is a cow $x$ such that any cow $y$ is equal to $x$" $\equiv$ "There exists $x$ such that $x$ is a cow and, for all $y$, if $y$ is a cow, then $x$ is equal to $y$" $\equiv \, \exists x, \, P(x) \wedge (\forall y, P(y) \to R(x,y))$
7. "There is a milk producing cow" $\equiv$ "There is $x$ such that $x$ is a cow and $x$ produces milk" $\equiv \ \exists x, P(x) \wedge Q(x)$
8. "Cows produce milk" $\equiv$ "Every cow produces milk" $\equiv$ "For all $x$, if $x$ is a cow, then $x$ produces milk" $\equiv \, \forall x, P(x) \to Q(x)$