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Prove that if A and B are sets, then (A - B) U B = A U B

  1. Oct 19, 2015 #1
    1. The problem statement, all variables and given/known data

    Prove that if A and B are sets, then (A - B) U B = A U B

    I think I might be missing a few steps here.

    2. Relevant equations


    3. The attempt at a solution

    (A - B) U B =

    1. (A ^ ~B) U B =

    2. (A ^ ~B) U (A ^ B) =

    3. A U B
     
  2. jcsd
  3. Oct 19, 2015 #2
    Hi Leo:

    I think that whether a step is "missing" or not depends on what your teacher expects.
    Hint:
    (W ^ X) U (Y ^ Z) = ((W U Y) ^ (X U Y)) U ((W U Z) ^ (X U Z))
     
  4. Oct 19, 2015 #3

    Krylov

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    Issue with step 1 to 2: When I'm in ##B##, but not in ##A##, I'm in set (1) but not in set (2).
    Issue with step 2 to 3: Set (2) is simply equal to ##A##, which may be smaller than set (3).

    I recommend you do not try to "rewrite" the sets, but instead prove two inclusions: First take an arbitrary ##x \in (A - B) \cup B## and argue step by step why it is then in ##A \cup B##. Then take an arbitrary ##x \in A \cup B## and argue why it is in ##(A - B) \cup B##.
     
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