Translating an English statement into a nested quantification with 3 variables.

Question

I am trying to translate:

"There is a student in this class who has been in every room of at least one building on campus."

My solution was:

$$ \exists x \exists b \forall r(B(x, b) \implies R(r, b)) $$

Where $B(x,b)$ represents "student x has been in building $b$", and $R(r, b)$ represents "room $r$ is in building $b$".

The correct solution is:

$$ \exists x \exists y \forall z(P(z, y) \implies Q(x, z)) $$

Where $P(z, y)$ represents "room $z$ is in building $y$", and $Q(x, z)$ represents "student $x$ has been in room $z$".

The way I understand the solution:

"There exists a student $x$ and there exists a building $y$ such that for all rooms $z$, if room $z$ is in building $y$ then student $x$ has been in room z."

Applying the same reading to my own answer, I am getting:

"There exists a student $x$ and there exists a building $b$ such that for all rooms $r$, if student $x$ has been in building $b$ then room $r$ is in building $b$."

So my answer is incorrect, because it is just saying all the rooms are in the building but not expressing the student has visited all the rooms?

Answer 1

Your solution permits for the possiblity that the student hasn't been anywhere. Then it still hold's that there is some building such that if the student was in that building, the room would be in there. It also entails that the building they visited, it any, contains all the rooms in existence in the universe of discourse, which is undesired.