Defintion of The Union Of Sets

  1. Hello,

    I am having trouble interpreting the definition of the union of two sets as given in Modern Abstract Algebra in Schaum's Outlines. I can see by example but I can't seem to interpret the definition. Could someone reword this for me or give me another spin on this definition? Thankyou.

    Defintion as in Modern Abstract Algebra in Schaum's Outlines

    Let A and B be given sets. The set of all elements which belong to A alone or to B alone or to both A and B is called the union of A and B.

    I understand the example:

    Let A = {1,2,3,4} and B = {2,3,5,8,10}; then A union B = {1,2,3,4,5,8,10}

    And the way I interpret the union of two sets is this:

    Given two sets A and B, let the union of A and B be C. Then C contains the following:

    Elements common to both A and B. Elements in A and not in B. And elements in B but not in A.


    But I don't get the definition as given by Schaums.

    Any help is appreciated. Thanks again.
     
  2. jcsd
  3. I don't really understand what the difference is between your definition and Schaums. You're both saying the same thing.
     
  4. I don't know. Perhaps I am misinterpreting Schuam's definition. I don't see the equivalence between Schaum's definition and mine.

    I interpret Schuams definition as follows:

    The set of all elements which belong to A alone. ==
    The set consisting of just the elements of A.

    Meaning if the set consists of just the elements of A then B is an empty set.

    A union B = A

    The set of all elements which belong to B alone ==
    The set consisting of just the elements of B.

    Similarly if the set consists of just the elements of B then A must be an empty set.

    B union A = B

    The set of all elements which belong to both A and B ==
    The set consisting of the elements in both A and in B.

    This last part I can see.
     
  5. u(AUB) = u(A) + u(B) - u(ANB) [The 'N' is intersect of A and B, in real life it looks like an upside-down U.]

    The two different cases are whether A and B are discrete sets or not. If A and B do not have anything in commmon, there is no intersection, hence u(ANB) = {} = 0, the empty set.
     
  6. Could you elaborate on your notation? I don't think I have seen the notation

    u(AUB)

    before.

    Just what does the u(...) stand for?

    I understand that if A and B have nothing in common that their intersection is the empty set.

    I don't see the connection though.
     
  7. u(AUB) represents the no. of elements in the set
     
  8. Right! Thanks.

    Now warr's post makes sense.
     
  9. It made sense before two, i was just giving him hints and i feel he has overlooked them
     
  10. Perhaps this can be of some assistance:
    [tex]A \cup B = \lbrace e \mid e \in A \lor e \in B \rbrace[/tex]

    The size of the union would be
    [tex]|A \cup B| = |A| + |B| - |A \cap B|[/tex]

    Whch means "all the elements in [tex]A[/tex] + all the elements in [tex]B[/tex] - the elements in both [tex]A[/tex] and [tex]B[/tex]".

    Nille
     
  11. You can easily see from attachment that Union means The no. of ekements in both sets without repeating the same no in both sets

    u can easily see Union will be RED+BLUE-GREY this is because as i say not to repeat the common elements we subtract the Grey portion once
    coz when we add RED+BLUE they common elements are added up twice so we have to delete one
     

    Attached Files:

  12. sumset

    a more general union is this. let T be a collection of sets, usually at least two sets. for example, T={A,B}.

    [tex]\bigcup T[/tex] is the set of elements in at least one member of T.

    [tex]\bigcup \left\{ A,B\right\} =A\cup B[/tex] is the set of elements in at least one of A and B. ie, if x is in A or B (or both), then x is in the union.

    if T had three sets in it, the same definition would apply: x is in the union of three sets if it is in at least one of the sets in T.
     
  13. Thanks again everyone.

    These many different perspectives has given me a better understanding of the def. of union of sets.
     
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