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Discrete math, proving the absorption law

  1. Apr 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove the second absorption law from Table 1 by showing
    that if A and B are sets, then A ∩ (A ∪ B) = A.

    2. Relevant equations
    Absorption laws
    A ∪ (A ∩ B) = A
    A ∩ (A ∪ B) = A


    3. The attempt at a solution
    i will show A ∩ (A ∪ B) is a subset of A
    x is any element in A ∩ (A ∪ B)
    x is not an element in (A ∩ (A ∪ B))'
    NOT ( x is an element in(A ∩ (A ∪ B))')
    NOT (x is not an element in A ∩ (A ∪ B))
    NOT (NOT (x is not an element in A ∩ (A ∪ B)))
    x is a element in A
     
  2. jcsd
  3. Apr 15, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    That's much too complicated. By definition of intersection, if x is in [itex]X\cap Y[/itex] then x is in both X and Y. So if x is in [itex]A\cap (A\cup B)[/itex] if follows immediately that x is in A.

    Of course to prove "X= Y" you must prove [itex]X\subset Y[/itex] and [itex]Y\subset X[/itex]. You have proved that [itex]A\cap(A\cup B)\subset A[/itex]. Now you must prove [itex]A\subset A\cap(A\cup B)[/itex]. Is x is in A then ....
     
  4. Apr 15, 2012 #3
    thank you for your reply
    so would the whole proof be

    1.A ∩ (A ∪ B) is a subset of A
    x is a element in A ∩ (A ∪ B)
    x is a element in A by definition of intersection
    Therefore A ∩ (A ∪ B) is a subset of A
    2.A is a subset of A ∩ (A ∪ B)
    x is a element in A
    x is a element in A ∩ (A ∪ B) by definition of intersection
    Therefore A is a subset of A ∩ (A ∪ B)
    3.Since A ∩ (A ∪ B) is a subset of A and A is a subset of A ∩ (A ∪ B),
    then A ∩ (A ∪ B) = A

    is the proof basically proofing they are subsets of each other by reversing each term?
     
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