I have 2 English sentences that I am having trouble translating them to predicate logic.
1. All people who live together drive the same car.
2. There is no number that is larger than all other numbers.
I do know that for sentence 1 it has to have a universal quantifier. The key word that gave it was All. Would this be correct or close to the answer ..
∀x(Person(x) ∧ Home(x) -> ∃y Car(y,x))
For the second one Same thing… Instead of a universal i use existential for the first part because it only refers to 1 number.
Can anyone help out with these? I'm just confused on how to tackle these. I'm looking for the answer but as well as the thoughts on how i can figure these out.
Best Answer
To simplify notation, use a two-sorted logic where lowercase letters are person variables and uppercase letters are car variables. Let $x\mathrm{T}y$ be the relation of $x$ and $y$ living together and $x\mathrm{D}A$ be the relation that $x$ drives $A$.
"All people who live together drive the same car." is equivalent to "For each person, there is a car that all people who live with him also drive." This is logically expressed as
$$ \forall x \exists A[x\mathrm{D}A \land \forall y[x\mathrm{T}y\rightarrow y\mathrm{D}A]]. $$
I'm not sure if it's intended that this car is the only car any of them drive, in which case the statement is $$ \forall x \exists A[x\mathrm{D}A \land \forall B[x\mathrm{D}B\rightarrow B=A]\land \forall y[x\mathrm{T}y\rightarrow (y\mathrm{D}A\land \forall C[y\mathrm{D}C\rightarrow C = A])]], $$ where $x\mathrm{D}A\land\forall B[x\mathrm{D}B\rightarrow B=A]$ expresses the idea that "the only car that $x$ drives is $A$".