Translate these statements into English, where R(x) is “x is a rabbit” and H(x) is “x hops”, and the domain consists of all animals.
$$(a) ∀xR(x) → H(x)$$
$$(b) ∀xR(x) ∧ H(x)$$
$$(c) ∃xR(x) → H(x)$$
$$(d) ∃xR(x) ∧ H(x)$$
The answers to these were the following:
(a) Every rabbit hops.
(b) All animals are rabbits and they all hop.
(c) There is an animal such that if it is a rabbit then it hops.
(d) There is a hopping rabbit.
Are my answers also correct?:
(a) For all x, if x is a rabbit, then x hops
(b) For all x, x is a rabbit and x hops
(c) There exists an x, if x is a rabbit then it hops
(d) There exists an x, where x is a rabbit and it hops
Edit: if they aren't correct, why not?