Statements to Predicates and Quantifiers

Question

"There is a student in this class who has taken every course offered by one of the departments in this school." I need to express this is predicates and quantifiers.

Can someone check my answer? I set $ x = student, y = course, z = department$. So I let $S(x,y)$ be "Student $x$ is in course $y$". And $T(y,z)$ be "Course $y$ is in department $z$".
Therefore: $\exists x \forall y \exists z S(x,y) \land T(y,z)$

Thanks,
Extreme112

Answer 1

The problem comes that $y$ could be a value thats not in the department, so we need an implication here. Let $U(x, y)$ denote that $x$ has taken $y$.

I would say, where $c$ is a particular class,

$\exists x : S(x, c) \land [\forall y : (\exists z : T(y,z)) \implies U(x, y)]$

That is, there exists $x$ such that $x$ is taking class $c$ forall $y$, if $y$ is offered by some department $z$, then $x$ is taking $y$.

Also note that if $x$ is taking class $c$ implies that $x$ has not taken $c$ then the answer is clearly "False".

Answer 2

"There is a student in this class who has taken every course offered by one of the departments in this school."

The particular entities are: this class $c$, this school $s$

The quantified entities are: one student $\exists x$, all courses $\forall y$, one department $\exists z$.

The predicates (or relations) are:

• $C(c,x):$ student $x$ is in this class $c$.

• $D(s,z):$ department $z$ belongs to this school $s$.

• $O(z,y):$ course $y$ is offered by department $z$.

• $T(x,y):$ course $y$ is taken by student $x$.

Then we translate the sentence as:

$$\exists x\, \Bigl(C(c,x)\wedge \exists z\, \bigl(D(s,z)\wedge \forall y\, (O(z,y)\to T(x,y))\bigr)\Bigr)$$

Note: This is interpreting the sentence to mean taking all of the courses that one (specific, unspecified) of the departments offers, rather than all of the courses that any of the departments offers.   The later would be:

$$\exists x\, \Bigl(C(c,x)\wedge \forall y\,\bigl(\exists z\, \bigl(D(s,z)\wedge O(z,y)\to T(x,y)\bigr)\bigr)\Bigr)$$

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cf "All items on one of the pages of the menu" vs "All items on any of the pages of the menu"

$\exists p\, \forall i\,( p\in m\wedge i\in p\to T(i) )$ vs $\forall i\, \exists p\, (p\in m\wedge i\in p\to T(i) )$