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Math Help - Prove By Distributive Law

  1. #1
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    Prove By Distributive Law

    Question, Prove by distributive law :

    a_{1} \cap ( a_2 \cup ... \cup a_n ) = ( a_1 \cap a_2 ) \cup ... \cup ( a_1 \cap a_n ) is true for all n>= 3

    any one got a clue on this problem? I am lost....
    Last edited by oldguy; November 22nd 2007 at 11:19 AM.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by oldguy View Post
    Question, Prove by distributive law :

    a_{1} \cap ( a_2 \cup ... \cup a_n ) = ( a_1 \cap a_2 ) \cup ... \cup ( a_1 \cap a_n ) is true for all n>= 3

    any one got a clue on this problem? I am lost....
    since you asked for a clue...use induction. the claim for n = 1 is by definition, use that to prove the claim true for n = 2, then use induction to show it's true for n >= 3
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  3. #3
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    Thanks for the clue. I will try your suggestion.
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  4. #4
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    Does any one know how to get started with this proble as Jhevon suggested?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by oldguy View Post
    Question, Prove by distributive law :

    a_{1} \cap ( a_2 \cup ... \cup a_n ) = ( a_1 \cap a_2 ) \cup ... \cup ( a_1 \cap a_n ) is true for all n>= 3

    any one got a clue on this problem? I am lost....
    Well, you know that
    a_1 \cap ( a_2 \cup a_3) = (a_1 \cap a_2) \cup (a_1 \cap a_3)

    So now consider the case for
    a_1 \cap ( a_2 \cup a_3 \cup a_4)

    Define a new set a_5 = a_3 \cup a_4.

    What is a_1 \cap ( a_2 \cup a_5)?

    -Dan
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