Reading Statements with Quantifiers I'm stuck on a question regarding quantifiers. It looks like this:
A(x,y) = x attended y's office hours
S = set of students (x)
T = set of teachers (y)

I have to write this in words:
¬∃ A(x, Professor Sandwich)


The answer to the problem is supposed to be: no one attended professor sandwiches office hours.
But I'm not understanding why it is read this way. I keep thinking it says: There exists no person x for which x attended Professor Sandwiches' office hours.
I think those answers say the same thing but my real question is this: Am I supposed to be plugging in the "there is no person x" into the function, leaving the "for which X" out? I don't understand how to read this nested quantifier statement. Any advice would be appreciated.
 A: Your statements say the same thing. I am no expert in the field of Mathematical Logic, but the statement - to me - is not very mathematical. It would have been better perhaps for the question to be phrased like $S$ is the set of students, $T$ is the set of professors and $A(t)$ is the set of students that attended Professors $t$'s office hours, where $t \in T$. Then to write $(\not\exists x \in S) \ni (x \in A(\text{Sandwich}))$. Then we can read this as there is no student (from the set of possible students) that is in the set of students that attend Professor Sandwich's office hours, i.e. no students attended Professor Sandwich's office hours. From this we know that $A(\text{Sandwich})= \emptyset$.
Ultimately, Mathematics, in this instance, is about communicating things (correctly). While I think the question is a bit ill-phrased, we still can still see what is meant (with a bit of work). $\not\exists x$ means there is no $x$, which we know comes from the set of students, so this is there is no students, and $A(x,y)$ is students $x$ that attend professor $y$ office hours. So we can see the statement implies there is no student, $x$, that we can evaluate in $A(x,y)$, meaning there is no student $x$ that attended professor $y$'s office hours. Now we just take $y=$ Professor Sandwich.
So you are correct, your statement just isn't very 'Englishy', in that its grammatically (mathematically) fine, but just isn't typically how we might phrase it to another person. Hence, the given answer over your answer appears as the 'desired' solution.
