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Help, set theory proving

  1. Nov 29, 2008 #1
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    I tried to do the questions but im just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks

    the symbol "n" means "intersect"
    U for Union


    1. The problem statement, all variables and given/known data
    1) Prove A contained B iff A n B = A

    2) Prove the following: For any sets A, B, C in a universe U:
    A n B = Universe iff A = Universe and B = Universe

    3) Prove or find counterexamples. For any sets A, B, C in a universe U:

    if A union C contained B union C then A contained B




    2. Relevant equations

    none.

    3. The attempt at a solution

    1) (=>) Assume A contained B

    Let x is an element of A, since A n A = A, x is an element of A and x is an element of B

    Case 1: x is an element of A: Since A contained B, x is an element of B so
    x is an element of A n B

    Case 2: x is an element of B: If x is an element of B then
    x is an element of (A n B)

    Hence x is an element of A n B

    This shows A contained A n B

    (<=) Assume A n B = A then

    A’=A’UA
    = A’ U (A n B)
    =(A’UA) n (A’U B)
    = empty set n A’ U B
    = A’ U B

    Hence
    Universe = A’ U B


    2) Suppose A n B = U and suppose that A is a proper subset of U then
    x is an element of B but
    x is not an element of A n B since x is not an element of A




    3) Let A be the empty set, and let B = C
    Then A union C = B and
    B union C = B so,
    A union C contains B union C, but A does not contain B because A is the empty set and B is not.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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