Kenneth Rosen Edition 6th Exercise 1.4 Question 11f (Page No. 59)

Question

Let S(x) be the predicate “x is a student,” F(x) the predicate
“x is a faculty member,” and A(x, y) the predicate
“x has asked y a question,” where the domain consists of
all people associated with your school. Use quantifiers to
express each of these statements.

Some student has not asked any faculty member a
question.

in rosen answer is given as-There is a student such that for every faculty member,

that student has not asked that faculty member a
question-----$\exists x(S(x) \Lambda \forall y(F(y)\rightarrow \sim A(x, y))).$

can i Write the above expression as-----$\forall y(F(y)\rightarrow \exists x(S(x) \Lambda \sim A(x, y))).$

for every faculty member, there exists some student who has not asked that faculty member a question.

is both the above statements have same meaning?

Comments

$\exists xS\left (x \right )\wedge$$\forall y \left (F\left ( y \right ) \rightarrow A\left ( x,y \right ) \right )$

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@srestha, 

$∃xS(x)∧∀y(F(y)→A(x,y))$ would mean "There is at least one student in the College and for all faculties in the College, a $specific$ person named, $x $ has asked them a Question"

(The Quantified Variable $x$ in your $\exists x  $ only has the Scope of $S(x)$, So, the $x$ in $A(x,y)$ doesn't have anything to do with the $x$ in $\exists x.$)

Frankly, $∃xS(x)∧∀y(F(y)→A(x,y))$ is Exactly Same as $∃zS(z)∧∀y(F(y)→A(x,y))$ which will be interpreted as "There is at least one student in the College and for all faculties in the College, a $specific$ person named, $x $ has asked them a Question"

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U mean without bracket

some student converted to only one student?

but why r u telling atleast?

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Proper placement of Brackets do matter.

Answer 1

$S(x) :$ $x$ is a Student.

$F(x) :$ $x$ is a Faculty.

$A(x,y ) :$ $x$ has asked $y$ a Question.

Let me consider that Domain of discourse, over which my variables $x,y$ may range,  consists of All People in a College.

Some student has asked every faculty member a question : 

We can re-write it as "There is some student $x$ such that If $y$ is a Faculty then $x$ has asked $y$ a Question. "

Which again can be re-written as "There is some  $x$ such that ($x$ is a Student and If $y$ is a Faculty then $x$ has asked $y$ a Question.) "

So, $\exists x(S(x) \wedge \forall y(F(y) \rightarrow A(x,y)))$

Comments

I was getting the same... but in Rosen solution is given- ∀y(F(y)→∃x(S(x) ∨ A(x,y)))

I don't understand the answer

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Given answer in Rosen is Not Correct.

Domain of discourse consists of All People in a College.

$T : ∀y(F(y)→∃x(S(x) ∨ A(x,y)))$ is True in Two cases :

Case 1 : There is at least one Student in the College.

Then  $T$ is Always True.

Case 2 : There is No Student in the College And For Every Faculty, there is some Person in the College who has asked that Faculty a Question.

$T_2 : ∀y(F(y)→∃x(S(x) \wedge A(x,y)))$

For Every Faculty, there is some Student in the College who has asked that Faculty a Question.

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can the statement be written as ∃x∀y(S(x)∧F(y)→A(x,y))

Answer 2

No. Both are different.

There is a student such that for every faculty member, that student has not asked that faculty member a
question

It is like "That student $x$ who doesn't ask any faculty(F1,F2,F3) a question". 

for every faculty member, there exists some student who has not asked that faculty member a question.

 It is like "For F1 there is a student $y$ who hasn't asked him a question, For $F2$ there is some(possibly some other) student $z$ who hasn't asked F2 any question.....etc"

Comments

@ Deepak sir
Is
First expression is talking about a unique student x who has not asked that faculty member a question?
But there can be more than 1 student like a and b
So why it is not like ---there exist some student (a or b)for every faculty member ,such that they have not asked that faculty member a question.

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First expression is talking about a unique student x who has not asked that faculty member a question?
But there can be more than 1 student like a and b 

No. It is asking for at least one such student. Say, You did not ask any question to any question. So, This statement is True for you.

 Second statement : for every faculty member, there exists some student who has not asked that faculty member a question.

Say, You did not ask any question to Arjun sir. But you have asked some question to Prashant sir. So, for Arjun sir, You are the student who hasn't asked him any question. So, this is like that.  

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Ok now, I got it :)