First Order Logic Question: English to predicate

Question

2. Let S(x) be the predicate "x is a student," B(x) the predicate "x is a book, " and H(x,y) the predicate "x has y, " where the universe of discourse is the universe, that is the set of all objects.  Use quantifiers to express each of the following statements.

    a) Every student has a book.

    b) Some student does not have any book.

    c) Some student has all the books.

    d) Not every student has a book.

    e) There is a book which every student has.

Answers: 

1. $\forall x (S(x) \implies \forall y (B(y) \wedge H(x,y)))$
2. $\forall x (S(x) \wedge \forall y (B(y) \implies \neg H(x,y)))$
3. $\forall x (S(x) \wedge \forall y (B(y) \implies H(x,y)))$
4. $\not \forall x (S(x) \implies \forall y (B(y) \wedge H(x,y)))$
5. $\forall x (B(x) \wedge \forall y (S(y) \implies H(x,y)))$

will someone explain  briefly ? I tried my best to dechiper in to predicate, But failed. 

Answer 1

This type of questions are somewhat tricky...Okk let starts with statment...

See every statement and see the beginning phrase properly..if it statemts with

1) Every or all =  2) There exist or Some =   3)Not every =  

No focus on each question implication is used whenever there you have to ensure that there is no condition such that left is true and rhs is false. and AND is used when both have to occur simultaneously...and note never assume anything out of question...

a) Every student has a book.

Means there will not be any one who is student and will not have means if he is student he has to have book means if Left(L) is true then that student have a book(Right Side ). Since there is "a book" means a single will do ur work. and since u cannot assume so if he is not a student then he may or may not have book .Thus,implication..Since student is main concern so x will be bound others..AND implies in right side that the object chosen y has be simultaneously a book and student must possess it. and since "a book" so one book will do it so there exist.

  x ( S(x)   y ( B(y)  H(x, y) ))

b) Some student does not have any book.

Or it can be read as there exist one student who does not have any book..Means You have to check entire book list and has to ensure that this student must not have any(simulatenously means he is student and check entire list that he does not have book).

  x ( S(x)   y ( B(y)   H(x, y) ))

e) There is a book which every student has.

Or All student have some particular book.....and this particular book can be any so  

 

 x B(x). and simulatenously you have to check that every student must possess that book so AND.in RHS we have implication which means there should be any student (Left=T) which do not have that book (Right=F)

 

 

 x ( B(x)   y ( S(y)  H(x, y) ))....

Try other one you will get it.....

I hope i m clear

Comments

if my answer for 

a) Every student has a book

 x y  S(x) B(y)     H(x, y)

is it wrong?

---

@ ezly007  I think the query implies that there is a book which every student has.