## A. Translation

Translate the following sentences into predicate logic notation.
Restrict yourself to the following vocabulary (and the equality
symbol):

```
Pxy = x is a parent of y
Lxy = x loves y
a = Alice
b = Bob
```

Assume that being a child means having a parent, being siblings means
having at least one parent in common, and being cousins means having at
least one grandparent in common. Assume also that quantifiers apply only
to people.

Everyone except Alice has a sibling.

Alice is Bob’s cousin.

Everybody loves their own grandchildren.

No person loves children unless they are a parent.

## B. Proofs and
Counterexamples

Consider the following two sentences:

∀xFx→P

∀x(Fx→P)

One of these two implies the other, but not vice versa. Give a proof
for the implication, and give a counterexample for the
non-implication.

Consider the following two sentences:

∀x∀y∃z(Rxz∧Ryz)

∃z∀xRxz

One of these two implies the other, but not vice versa. Give a proof
for the implication, and give a counterexample for the
non-implication.

Could there be a correctly written proof with the following line
fragments? Explain your answer with reference to soundness or
completeness.

```
1 (1) ∃x(Fx∧Gx) A
2 (2) ∃x(Gx∧Hx) A
1,2 (n) ∃x(Fx∧Hx)
```

Could there be a correctly written proof with the following line
fragments? Explain your answer with reference to soundness or
completeness. (Hint: ∃x∀y(Fx→Fy) is logically equivalent to
∃x(Fx→∀yFy).)

```
1 (1) ¬∃x∀y(Fx→Fy)
1 (n) ∀x(Fx↔¬Fx)
```