Converting statements with quantifiers

Question

I'm having a little trouble understanding quantifiers and therefore doubting all my study answers. Since there is no where to check if the answers are correct, I'm very very worried I am just practicing incorrectly. So I've set up two examples with what I think the answers are.

It would be brilliant if you could confirm if I am correct or not so I could use these answers as a base to check my other answers. If I am incorrect, it would be awesome if you could point me the right direction!

Example 1: Nobody except Jay and Mike is walking home.

My answer: ∀x [walking home(x) ∧ [¬ mike(x) ∧ ¬ Jay(x)]]

Example 2: There is a time to watch, and a time to bluff; a time to fold, and a time to check.

My answer: ∀x,y,z,v [watch(x) ∧ bluff(y) ∧ fold (z) ∧ check(v)] I chose ∀ since I've always seen it as another way of saying "always" but I could be wrong.

Answer 1

Nobody is walking home.

$M(x):$ "x is Mike".
$J(x):$ "x is Jay."
$W(x):$ "x is walking home."

$$\lnot \exists x\,\Big(\lnot M(x) \land \lnot J(x) \land W(x)\Big) \equiv \forall x\,\Big((\lnot M(x) \land \lnot J(x)) \implies \lnot W(x)\Big)$$

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In the second translation, each and every $\forall$ symbol needs to be an $\exists$ symbol, and I am assuming $x, y, z, v$ all belong to the universe defined by "times".

"There is a time for P" is just like saying: "There exists a time for P" which translates to $\exists x P(x)$. But the connective $\land$ you use is correct.

$W(x):$ "x is a time to watch."
$B(x):$ "x is a time to bluff."
$F(x): $ "x is a time to fold."
$C(x): $ "x is a time to check."

$$\exists x,y,z,v \,\Big(W(x) \land B(y) \land F(z) \land C(v)\Big)$$

Answer 2

Example 1: So here x is a person, walking home(x) is true if x is walking home.

∀x [walking home(x) ∧ [¬ walking home(mike) ∧ ¬ walking home(Jay)]

Example 2: Since the example says "there is a time" you should use ∃.

∃x,y,z,v [watch(x) ∧ bluff(y) ∧ fold (z) ∧ check(v)] 

Answer 3

Nobody except Jay and Mike is walking home.

We would normally translate names with names (constants), not predicates. Thus, assuming the domain of quantification is people, and using '$j$' as a constant to denote Jay and '$m$' as a constant to denote Mike, then

$$\forall x(Wx \to (x = j \lor x= m))$$

would be neat -- take anyone you like, if they are walking home then they are either Jay or Mike. Or equivalently, but perhaps somewhat less prettily, we could have

$$\neg\exists x(Wx \land (\neg x = j \land \neg x = m))$$

For the other question, as others explain, you need an existential quantifier to render the natural reading of 'There is a time ...'.