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Discrete Mathematics - Basic Set Theory : Assignment review : Q2

  1. Jul 16, 2012 #1
    Question 2:
    --------------------

    1. The problem statement, all variables and given/known data

    Consider the following sets, where U represents a universal set :

    U = {1, 2, 3, 4, ∅, {1}}
    A = {1, 3}
    B = {{1}, 1}
    C = {2 , 4}
    D = { ∅ , 1, 2 }

    2. Relevant equations

    A+D is the set : (Choose only one )

    1. {1, 3}
    2. {1, 2, 3}
    3. {∅, 2, 3}
    4. {2, 3}


    3. The attempt at a solution

    This is quite an easy one, however i am a bit confused - will explain now.
    The answer , to my knowledge, is number 3.

    If the 2 sets are combined with '+', the sets are added together, however all duplicates are removed. when all duplicates are removed , we are left with the following :

    A + D = { ∅, 2, 3}

    Please confirm if the above is correct.

    However, the question I have :

    If ∅ is included in every set, then why do we have to include it in this set, and additionally, if it is in every set, will it not be that it must be removed from A + D, so that we end with the set A+ D = { 2, 3 }

    I am a bit confused with ∅ being included in every set.

    Please be so kind as to help me with this one and advise where I am making an obvious error.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 16, 2012 #2

    SammyS

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    ∅ = {} is the empty set, also known as the null set. It is not a member of every set, it is a subset of every set.

    In this problem, the set D has {}, a.k.a. ∅, as one of its elements, just as set B has the set {1} as one of its elements.

    For the sets in this problem the set, ∅, is a subset of all of them. However, the set, ∅, is an element of only sets, U and D. Also, the set {∅}, a.k.a. {{}}, is a subset of sets, U and D, but no others.
     
  4. Jul 21, 2012 #3
    Thanks for the reply, so the correct answer , would be number 3.

    Please advise if this is correct, as this is what I understand from the set having {} as a sub set, however not {} as an element.

    Thanks again for helping me.
     
  5. Jul 21, 2012 #4

    HallsofIvy

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    I don't believe that "+" is standard notation for an operation in basic set theory. You appear to be using it to mean the symmetric difference , more commonly, I believe, given as [itex]A\Delta B[/itex]. Is that correct?

    If so then, yes, the symmetric difference of A and D is {∅, 2, 3}. Yes, ∅ is a subset of every set. It is a member of a set only if it is specifically given as such, as in "D".
     
    Last edited: Jul 21, 2012
  6. Jul 21, 2012 #5
    Hi ... yes, you are correct. I have not heard of 'symmetric differences' before, and had to google it. http://en.wikipedia.org/wiki/Symmetric_difference is what I found.

    So to me the following is thus the same :

    + and your sign means the same.

    what is the international convention ?
     
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