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Math Help - Symbolic Form help

  1. #1
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    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: \exists b, B(p,b)

    b) Book b has been borrowed.

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

    c) Book b is on the shelf

    My Answer: Not sure

    d) Person p has borrowed at least two books.


    My Answer: Not sure

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

    My Answer: Not sure.

    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.
    Last edited by jvignacio; April 10th 2010 at 02:03 AM.
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  2. #2
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    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.)
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  3. #3
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    Quote Originally Posted by emakarov View Post
    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) \sim B(b)

    and im not sure how to write d) and e) any help?
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  4. #4
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    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 B takes two arguments. Since " b has been borrowed" is written correctly in b), how would one write "it is not the case that 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., \exists c,\;\dots or \exists x,\;\dots.

    Next, "There exists two non-equal books such that..." means "There exists a book b_1 such that there exists a book b_2 such that b_1 does not equal b_2 and ...". Finally, "the person p borrowed one book and the other" means "the person p borrowed one book and 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 \exists b,\;\dots. Of course, one needs to know very well what English phrases are denoted by \exists, \forall, \land, \to, etc.
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  5. #5
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    Quote Originally Posted by emakarov View Post
    The predicate B takes two arguments. Since " b has been borrowed" is written correctly in b), how would one write "it is not the case that b has been borrowed"?
    Would it be \sim B(p,b) ?
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  6. #6
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    Quote Originally Posted by emakarov View Post
    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: \exists b_1, \exists b_2, B(p, b_1) \wedge B(p, b_2) ?
    Last edited by jvignacio; April 12th 2010 at 07:36 AM.
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  7. #7
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    Anyone know? Cheers
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  8. #8
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    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 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 {}\land b_1\ne b_2 because d) says "at least two books", and in saying \exists b_1\,\exists b_2 nothing prevents b_1 and b_2 to be equal.
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  9. #9
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    Quote Originally Posted by emakarov View Post
    You only need to add {}\land b_1\ne b_2
    Sorry what am I adding this too? Im not sure what you mean.
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  10. #10
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    To the conjunction B(p,b_1)\land B(p,b_2).
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  11. #11
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    Quote Originally Posted by emakarov View Post
    To the conjunction B(p,b_1)\land B(p,b_2).
    Ok thank you emakarov. Much appreciated.
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  12. #12
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    Quote Originally Posted by emakarov View Post
    To the conjunction B(p,b_1)\land B(p,b_2).
    Also for:

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

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