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Set theory proof

  1. Oct 3, 2016 #1

    Math_QED

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    1. The problem statement, all variables and given/known data

    Prove that ##A \cup (A \cap B) = A##

    2. Relevant equations

    In the previous exercise, we proved:

    Let A, B be sets. Then, the following statements are equivalent:

    1) ##A \subseteq B##
    2) ##A \cup B = B##
    3) ##A \cap B = A##

    3. The attempt at a solution

    The proof of ##A \cup (A \cap B) = A## according to the teacher was: we can use this previous exercise to show that ##A \cap B = A## Then the problem becomes that ##A \cup (A \cap B) = A \cap A = A##

    However, we are not given that ##A \subseteq B##. Hence, we cannot apply this previous exercise. So, is the teacher's proof wrong?

    Btw, can someone verify my proof:

    Proof:

    To show that ##A \cup (A \cap B) = A##, we show that ##A \cup (A \cap B) \subseteq A \land A \subseteq A \cup (A \cap B)##

    1) Take ##x \in A \Rightarrow x\in A \cup (A \cap B)## by definition of union.
    We deduce that ##A \subseteq A \cup (A \cap B)##

    2) Take ##x \in A \cup (A \cap B) \Rightarrow x \in A \lor x \in A \cap B##
    ##\Rightarrow x \in A \lor (x \in A \land x \in B)##
    ##\Rightarrow x \in A## (logic argument: ##p \lor (p \land q) \Rightarrow p##)
    We deduce that ##A \cup (A \cap B) \subseteq A##

    QED.
     
    Last edited: Oct 3, 2016
  2. jcsd
  3. Oct 3, 2016 #2

    PeroK

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    What do you think?
     
  4. Oct 3, 2016 #3

    Math_QED

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    I think he's wrong. Can you verify the proof that I made please? (I edited my post)
     
  5. Oct 3, 2016 #4

    PeroK

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    I think your teacher might have meant ##A \cap B \subseteq A## hence ##A \cup (A \cap B) = A##

    Your proof looks a bit over-elaborate to me.
     
  6. Oct 3, 2016 #5

    Math_QED

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    I'm not sure what the teacher meant. Next time I'll see him I'll ask what he meant. Thanks for your help.
     
  7. Oct 3, 2016 #6

    SammyS

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    You do know that ##A \cap B\subseteq A\,,\ ## Right?

    The proof pretty much follows from there.
     
  8. Oct 4, 2016 #7

    Math_QED

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    Wow didn't see that. Guess that happens when it's late. Thanks!
     
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