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Subset Question

  1. Jan 12, 2015 #1
    1. The problem statement, all variables and given/known data
    I am trying to prove the absorption law
    A U (A ∩ B) = A
    I know that a way to prove this is to show that each is a subset of the other but I'm a little confused about one part in the process (below)

    2. Relevant equations


    3. The attempt at a solution
    Let x∈A U (A ∩ B)
    then x∈A or x∈(A ∩ B)
    so because x∈A then we know that A U (A ∩ B) ⊆ A (I don't understand why this line is true.)

    Why just because x∈A does it mean that A U (A ∩ B) ⊆ A is true? Any help is greatly appreciated.
     
  2. jcsd
  3. Jan 12, 2015 #2

    haruspex

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    Arguably there's a step missing in there.
    If x∈(A ∩ B) then x∈A , so either way x∈A U (A ∩ B) implies x∈A.
    Thus you have shown that every element of A U (A ∩ B) is an element of A. Hence A U (A ∩ B) ⊆ A.
     
  4. Jan 12, 2015 #3
    This is true because each element in the subset ' A U (A ∩ B) ' must belong to A .
     
  5. Jan 13, 2015 #4

    RUber

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    This is a logical argument. You are trying to show that if x is in ##A \cup ( A \cap B)##, then it is also in A, and if x is in A, then it is in ##A \cup ( A \cap B)##.
    You have already shown the first part (edit) by the definition of the intersection: if x is in ##A \cup ( A \cap B)##, then it is also in A, which implies that ##A \cup ( A \cap B)\subseteq A ##,
    Next, you need to show that ##A \subseteq A \cup ( A \cap B) ##. That should be simple enough by the definition of a union. So it looks like you are just about done.
     
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