1. Any apartment in London has lower rent than some apartments in Paris.

$$ \forall {x} [{Apt}(x) \land {In}(x,{London})] \implies \exists {y} ([{Apt}(y) \land {In}(y,{Paris})] \implies ({Rent}(x) < {Rent}(y))) $$ 2. There is exactly one apartment in Paris with rent below \$1000.

$$ \exists {x} {Apt}(x) \land {In}(x,{Paris}) \land \forall{y} [{Apt}(y) \land {In}(y,{Paris}) \land ({Rent}(y) < {Dollars}(1000))] \implies (y = x) $$ 3. If an apartment is more expensive than all apartments in London, it must be in Moscow.

$$ \forall{x} {Apt}(x) \land [\forall{y} {Apt}(y) \land {In}(y,{London}) \land ({Rent}(x) > {Rent}(y))] \implies {In}(x,{Moscow}). $$

For each of the following sentences in English, decide if the
accompanying first-order logic sentence is a good translation. If not,
explain why not and correct it.

1. Any apartment in London has lower rent than some apartments
in Paris.

$$
\forall {x} [{Apt}(x) \land {In}(x,{London})]
\implies \exists {y} ([{Apt}(y) \land {In}(y,{Paris})] \implies ({Rent}(x) < {Rent}(y)))
$$
2. There is exactly one apartment in Paris with rent below \$1000.

$$
\exists {x} {Apt}(x) \land {In}(x,{Paris}) \land \forall{y} [{Apt}(y) \land {In}(y,{Paris}) \land ({Rent}(y) < {Dollars}(1000))] \implies (y = x)
$$
3. If an apartment is more expensive than all apartments in London, it
must be in Moscow.

$$
\forall{x} {Apt}(x) \land [\forall{y} {Apt}(y) \land {In}(y,{London}) \land ({Rent}(x) > {Rent}(y))] \implies
{In}(x,{Moscow}).
$$