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Math Help - Sum Rule question

  1. #1
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    Sum Rule question

    Hello,

    I found this problem in one textbook and it's driving me nuts:

    An instructor has two colleagues. One has three textbooks on analysis of algorithms, and the other has five such textbooks. If n denotes the maximum number of different books on this topic, then 5 <= n <= 8 is what can be borrowed from these two instructors.

    My problem is, shouldn't it be 3 <= n <= 8?

    Could somebody please explain to me why it is 5 <= n <= 8 if it's correct?

    Thanks,
    Victor.
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  2. #2
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    Quote Originally Posted by victorsk View Post
    Hello,

    I found this problem in one textbook and it's driving me nuts:

    An instructor has two colleagues. One has three textbooks on analysis of algorithms, and the other has five such textbooks. If n denotes the maximum number of different books on this topic, then 5 <= n <= 8 is what can be borrowed from these two instructors.

    My problem is, shouldn't it be 3 <= n <= 8?

    Could somebody please explain to me why it is 5 <= n <= 8 if it's correct?

    Thanks,
    Victor.
    I assume one instructor has three DIFFERENT textbooks and the other instructor has five DIFFERENT textbooks. If the second instructor has five DIFFERENT textbooks, then the maximum must be at least five, hence the answer being in the range of n: [5,8]

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  3. #3
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    Hi,

    Thanks for replying. It says "both colleagues may own copies of the same textbook(s)".

    My apologies for being a bit thick but I am trying to understand why the maximum has to be at least 5?

    I never get these problems and have to take Discrete Mathematics this fall so I start early to study it.


    Thank you,
    Victor.
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  4. #4
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    Quote Originally Posted by victorsk View Post
    Hi,

    Thanks for replying. It says "both colleagues may own copies of the same textbook(s)".

    My apologies for being a bit thick but I am trying to understand why the maximum has to be at least 5?

    I never get these problems and have to take Discrete Mathematics this fall so I start early to study it.


    Thank you,
    Victor.
    Well, both colleagues may own the same text, but if one colleague owns five different textbooks, say A, B, C, D and E, then the other colleague can own three textbooks.

    These three textbooks may be three of the five that the other colleague has. If that is the case, then the total number of unique textbooks is 5. However, the other instructor can have a completely different set of textbooks, say F, G and H. This give us a grand total of 8.

    You cannot go below 5 or higher than 8.
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  5. #5
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    So, if first instructor has A, B, C, D, E and the second instructor has, say, A, B, C, then I can see how you can't go below 5 as the lowest number (because the first instructor is the "superset" of the books for both instructors?) and so by the sum their total number of books is 8.

    But if the other instructor has F, G, and H, isn't the lowest number of books in this case is 3 because they are not among the 5 books of the first instructor?

    Thanks,
    Victor.
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  6. #6
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    Quote Originally Posted by victorsk View Post
    But if the other instructor has F, G, and H, isn't the lowest number of books in this case is 3 because they are not among the 5 books of the first instructor?

    Thanks,
    Victor.
    This is where your logic went wrong. What is the question asking for?

    If n denotes the maximum number of different books on this topic
    This is amongst both of the instructors, so if all of their books are different, then there will be 8 total unique books. I hope this has helped!
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  7. #7
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    Hi,

    Sorry, so is the number 5 then a minimum among the maximum number of unique books, i.e. are we taking the maximum "minimum" number of unique books among both instructors, which is 5?

    Thanks,
    Victor.
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  8. #8
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    Quote Originally Posted by victorsk View Post
    Hi,

    Sorry, so is the number 5 then a minimum among the maximum number of unique books, i.e. are we taking the maximum "minimum" number of unique books among both instructors, which is 5?

    Thanks,
    Victor.
    Five is the minimum number of books.

    Eight is the maximum number of books.
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  9. #9
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    Hi,

    I'm still trying to understand this. Can I alter the problem slightly so I can get it?

    Suppose a student has a choice between 40 books on math and 50 books on sociology. Will the total number of books that he can borrow be defined

    n : [40, 90] ??

    Thanks,
    Victor.
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