Exercise 8.29

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.

$$\forall {x} [{Apt}(x) \land {In}(x,{London})] \implies \exists {y} ([{Apt}(y) \land {In}(y,{Paris})] \implies ({Rent}(x) < {Rent}(y)))$$

1. There is exactly one apartment in Paris with rent below $1000.

$$\exists {x} {Apt}(x) \land {In}(x,{Paris}) \land \forall{y} [{Apt}(y) \land {In}(y,{Paris}) \land ({Rent}(y) < {Dollars}(1000))] \implies (y = x)$$

1. If an apartment is more expensive than all apartments in London, it must be in Moscow.

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