1. Symbolic Form help

Hey guys, question regarding symbolic form.

In the design specification of a library system, B(p,b) denotes the predicate 'person p has borrowed b', and O(b) denotes the predicate 'book b is overdue'.
Write the following sentences in the symbolic form:

a) Person p has borrowed a book.

My Answer: $\displaystyle \exists b, B(p,b)$

b) Book b has been borrowed.

My Answer: $\displaystyle \exists p, B(p,b)$

c) Book b is on the shelf

d) Person p has borrowed at least two books.

e) No book has been borrowed by more than one person.

If you guys could let me know if my answers are correct and help me with the ones i dont know id highly appreciate it. Thanks.

2. c) Book b is on the shelf
It is not the case that b has been borrowed. (Provided it has not been lost :-)

d) Person p has borrowed at least two books.
There exists a person and two non-equal books such that the person borrowed one and the other.

e) No book has been borrowed by more than one person.
For any book and two non-equal persons, if they borrowed this book, then False (contradiction, 0 = 1, p not equal to p, etc.)

3. Originally Posted by emakarov
It is not the case that b has been borrowed. (Provided it has not been lost :-)

There exists a person and two non-equal books such that the person borrowed one and the other.

For any book and two non-equal persons, if they borrowed this book, then False (contradiction, 0 = 1, p not equal to p, etc.)
Ok so:

c) $\displaystyle \sim B(b)$

and im not sure how to write d) and e) any help?

4. First, a correction: in d), one does not have to quantify over people because the person p, who borrowed at least two books, is given explicitly. So, the more formal English variant is "There exist two non-equal books such that p borrowed one book and the other book".

c)
The predicate $\displaystyle B$ takes two arguments. Since "$\displaystyle b$ has been borrowed" is written correctly in b), how would one write "it is not the case that $\displaystyle b$ has been borrowed"?

and im not sure how to write d)
General remarks. "There exist two dogs such that ..." is a contraction for "There exists a dog such that there exists a dog such that..." Next, when translating "there exists a cat such that" one gives some temporary name to the cat; e.g., $\displaystyle \exists c,\;\dots$ or $\displaystyle \exists x,\;\dots$.

Next, "There exists two non-equal books such that..." means "There exists a book $\displaystyle b_1$ such that there exists a book $\displaystyle b_2$ such that $\displaystyle b_1$ does not equal $\displaystyle b_2$ and ...". Finally, "the person $\displaystyle p$ borrowed one book and the other" means "the person $\displaystyle p$ borrowed one book and $\displaystyle p$ borrowed the other book".

Expanding the phrase in such way, one only needs to substitute symbolic expressions for English words. E.g., "There exists a book such that ..." is replaced by $\displaystyle \exists b,\;\dots$. Of course, one needs to know very well what English phrases are denoted by $\displaystyle \exists$, $\displaystyle \forall$, $\displaystyle \land$, $\displaystyle \to$, etc.

5. Originally Posted by emakarov
The predicate $\displaystyle B$ takes two arguments. Since "$\displaystyle b$ has been borrowed" is written correctly in b), how would one write "it is not the case that $\displaystyle b$ has been borrowed"?
Would it be $\displaystyle \sim B(p,b)$ ?

6. Originally Posted by emakarov
First, a correction: in d), one does not have to quantify over people because the person p, who borrowed at least two books, is given explicitly. So, the more formal English variant is "There exist two non-equal books such that p borrowed one book and the other book".
Would this be: $\displaystyle \exists b_1, \exists b_2, B(p, b_1) \wedge B(p, b_2)$ ?

7. Anyone know? Cheers

8. Quote:
Originally Posted by emakarov
The predicate takes two arguments. Since " has been borrowed" is written correctly in b), how would one write "it is not the case that has been borrowed"?

Would it be ?
No, the English statement in c) does not say anything about the particular person $\displaystyle b$, like your version does. The answer is obtained by adding ~ in front of the answer to b).

Would this be: ?
You only need to add $\displaystyle {}\land b_1\ne b_2$ because d) says "at least two books", and in saying $\displaystyle \exists b_1\,\exists b_2$ nothing prevents $\displaystyle b_1$ and $\displaystyle b_2$ to be equal.

9. Originally Posted by emakarov
You only need to add $\displaystyle {}\land b_1\ne b_2$
Sorry what am I adding this too? Im not sure what you mean.

10. To the conjunction $\displaystyle B(p,b_1)\land B(p,b_2)$.

11. Originally Posted by emakarov
To the conjunction $\displaystyle B(p,b_1)\land B(p,b_2)$.
Ok thank you emakarov. Much appreciated.

12. Originally Posted by emakarov
To the conjunction $\displaystyle B(p,b_1)\land B(p,b_2)$.
Also for:

e) No book has been borrowed by more than one person.

is this:$\displaystyle \forall b, \exists p_1, \exists p_2, B(p_1, b) \wedge B(p_2, b) \wedge p_1 \neq p_2$ ?

and

f) there are no overdue books

is this: $\displaystyle \forall b, \sim O(b)$ ? -> Note: Book b is overdue = $\displaystyle O(b)$