Use quantifiers and predicates with more than one variable to express this statement:
There is a student in this class who has taken every course offered by one of the departments in this school
I know there are many ways we can express a single statement with logical symbols but I'd like to know if the solution I came up with needs to be corrected, because sometimes I am confused dealing with the domain of the variable
My solution:
T(x, y) denotes "x have taken y" where the domain of x consists only of students and y is a course
O(y, z) denotes "y is a course of the department z" where z is a department of the school
∃x ∀y ∃z (T(x, y) ∧ O(y, z))
Best Answer
The statement
could be "worded" more clearly. I take it to mean that there is some student $x$ and there is some ("one") department $z$, such that $x$ has taken every course $y$ offered by $z$: $$\exists x\,\exists z\,\forall y\,(O(y,z) \to T(x,y)). \tag{1} $$ Notice that in the highlighted English sentence, the "every course" quantifier precedes the department quantifier ("one of the departments"), the opposite of their order in the logical structure.
You didn't symbolize "in this class", so I didn't either in (1), but you should. To keep it simple, let $C(x)$" stand for "$x$ is in this class". A better rendering: $$\exists x\, \big( C(x) \land \exists z\,\forall y\,(O(y,z) \to T(x,y)) \big). \tag{2} $$ The "domain" over which variables range is a little unusual for a basic logic example but quite typical of "real-world" situations. It's most naturally thought of as multi-sorted: there are students, departments, and courses, so the actual universe of the interpretation is the disjoint union of these sets. In the intended interpretation of $T$, if $T(x,y)[\mathbf{a}/x, \mathbf{b}/y ]$ is true, then $\mathbf{a}$ is a student and $\mathbf{b}$ is a course; similarly, $O(y,z)[\mathbf{b}/y, \mathbf{c}/z ]$ is true only of courses $\mathbf{b}$ and departments $\mathbf{c}$.
Your statement is very different than what the English sentence (tries to) express. Your rendering: $$\exists x\,\forall y\,\exists z\,(T(x,y) \land O(y,z)) $$ is equivalent to: $$\exists x\,\forall y\,T(x,y) \land \forall y\,\exists z\,O(y,z) $$ which says: there's a student who has taken every course, and every course is offered by some department — two independent statements.