I've seen several other questions on here pertaining to this subject, but i'm honestly struggling to apply the answers from those posts to my own issue. I am confused on how to use the predicates to express the statements. See my question and answers below (I apologize I have a feeling they're all wrong).
Question:
Let: S(x) be the predicate "x is an employee"
F(x) the predicate "x is a manager"
A(x,y) the predicate "x has asked y a question"
Where the domain consists of all members of your company
Use quantifiers to express each of these statements
a) robert has asked Managing Director Michael a question.
b) every employee has asked Manager paul a question.
c) Some employee has not asked any manager a question.
d) There is an manager who has never been asked a question by a student.
e) Some employee has asked every manager a question.
f) Some employee has never been asked a question by a manager.
My Answers
a) $\exists$x $\exists$y A(x,y)
b) $\forall$x $\exists$y A(x,y)
c) $\exists$x $\forall$y A(x,y)
d) $\forall$x $\exists$y A(x,y)
e) $\exists$x $\forall$y A(x,y)
f) $\forall$x $\exists$y A(x,y)
Updated Answer Attempts
a) A(l,m) $\land$ F(m)
b) $\forall$x(S(x)) $\rightarrow$ A(x,p)
c) $\exists$(S(x) $\land$ $\lnot$ $\exists$y(F(y) $\land$ A(x,y))
d) $\exists$(F(y) $\land$ $\lnot$ $\exists$x(S(x) $\land$ A(x,y))
e) $\exists$ (S(x) $\land$ $\forall$y(F(y)) $\land$ A(x,y)
f) $\exists$(S(x)) $\land$ $\forall$ (F(y)) $\land$ $\lnot$ A(y,x)
Best Answer
Couple of comments:
First, want to use specific constants, rather than variables, to refer to specific individuals.
So, for example, for your first sentence use $A(r,m)$, with the understanding that you use $r$ to denote Robert, and $m$ for Michael.
Second, make sure to use the predicates of being an employee or manager where it says so in the sentence. So, for example, for your first sentence we are told Michael is a manager, so add $F(m)$ to your sentence.
Thus, the first sentence becomes:
$A(r,m) \land F(m)$
Third, sentences of the form "All p's are q's" get translated to $\forall x (p(x) \to q(x))$, while 'Some p's are q's" become $\exists x (p(x) \land q(x))$
Thus, for example, your second sentence becomes:
$\forall x (S(x) \to A(x,p))$
And your third:
$\exists x (S(x) \land \neg \exists y (F(y) \land A(x,y)))$
Ok: you try again the other three!