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Math Help - Set Help

  1. #1
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    Set Help

    Currently doing a short maths course to aid and help in 3d modeling and animation,just wondering if anyone could help in set theory?

    The questions were as follows and my answers too.

    [A]
    Given the following sets: r =the set of registration numbers of all vehi-
    cles (i.e. Consider
    r as the type or Universe for this problem), c =the set of
    car registration numbers,
    m =the set of motorcycle registration numbers,
    and
    V =the set of registration numbers of old Vehicles

    Describe the following in words e.g. m (intersection) V = the set of all old motorcylces.
    (i) (V(union)m);
    The set of old vehicles and all registration numbers
    (ii) (V (intersection) (cnr))(symmetric difference) c;
    The set of car registration numbers
    Write down the following sets using set notation
    (iii) the set of new vehicles that aren't a motorcycle;
    (r(symmetric difference)c)(union)V
    (iv) the set of all vehicles that are a car or are an old motorcycle
    (c\r)(union)(m(intersection)V)

    [B]
    Considering any 4 subsets of a universe, U, work out (and show your
    working)
    (v) how many different regions there could be.
    8 different regions
    (vi) how many dffierent ways of combining these regions using boolean combinations are there?
    256 of combining these

    [C]
    (vii) Decide and state whether any subset of a countable set is itself

    countable
    . Explain your answer in your own words.


    Thats it guys, sorry to drag it out also Rather then using set notation I have used words in brackets cause everytime I tried to copy and paste it came out with something completly different.

    Thnaks in advance

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  2. #2
    MHF Contributor
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    (i) (V(union)m);
    The set of old vehicles and all registration numbers
    The set m is the set of motorcycle registration numbers.

    (ii) (V (intersection) (cnr))(symmetric difference) c;
    The set of car registration numbers
    Does n mean intersection in "cnr"? I.e., is the expression (V\cap(c\cap r))\vartriangle c? (By the way, why is V an uppercase letter while all others, including the universal set, are lowercase?) In this case, c\cap r = c and V\cap c is the set of old cars. Further, V\cap c\subseteq c, so (V\cap c)\vartriangle c=c\setminus V.

    (iii) the set of new vehicles that aren't a motorcycle;
    (r(symmetric difference)c)(union)V
    This answer makes little sense. First, the question does not even refer to cars. If you use r\vartriangle c (which is equal to r\setminus c) to exclude cars so that only motorcycles remain, this may not work unless you are told that r=c\cup m. For example, there may be trucks, which are neither cars nor motorcycles. Then, it is the motorcycles that you need to exclude. Also, you need to exclude V, while you are taking a union with V.

    (iv) the set of all vehicles that are a car or are an old motorcycle
    (c\r)(union)(m(intersection)V)
    c\setminus r is the empty set because c\subseteq r. The answer is c\cup (m\cap V).

    Considering any 4 subsets of a universe, U, work out (and show your
    working)
    (v) how many different regions there could be.
    8 different regions
    It is not clear what a region is. If you mean the number of disjoint areas in a Venn diagram, then for n sets, there are 2^n areas.

    (vi) how many dffierent ways of combining these regions using boolean combinations are there?
    256 of combining these
    The question is not clear to me. In particular, it is not clear if more than two regions can be combined.
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  3. #3
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    sorry, (V (intersection) (cnr))(symmetric difference) c;
    was meant to be (V (intersection) (c\r))(symmetric difference) c;

    Thanks for the swift response

    JT.
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  4. #4
    MHF Contributor
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    Since c\subseteq r, c\setminus r=\emptyset, V\cap(c\setminus r)=\emptyset and \emptyset\vartriangle c=c, so yes, this is the set of car registration numbers.
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  5. #5
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    Feb 2011
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    Thanks again for the swift response, first time I have delved into sets , it is just taking a little time to sink in.

    Thanks again
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