Results 1 to 3 of 3

Math Help - Proof

  1. #1
    Newbie
    Joined
    Dec 2006
    Posts
    23

    Proof

    Hi guys
    Could anybody help me with the proof
    Proof-picture-003.jpg
    Appreipriate it
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,902
    Thanks
    329
    Awards
    1
    Quote Originally Posted by mauro21pl View Post
    Hi guys
    Could anybody help me with the proof
    Click image for larger version. 

Name:	Picture 003.jpg 
Views:	26 
Size:	20.7 KB 
ID:	1668
    Appreipriate it
    I don't have my book with me at the moment (and I don't trust myself to do the full proof without it, no matter how elementary), so I won't do the whole thing, but think about this as an induction proof. De Morgan's Law holds for two sets. Can you think of a way to show that if it holds for N sets that it will also hold for N+1? (Given that it holds for 2 sets.)

    -Dan

    Note: I'm having some thoughts about this for N \to \infty. I don't know if you might not run into trouble with that. But as your problem specifies an "n" as an upper index you should be fine for your proof.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    There are two basic ways to prove this.
    First by DeMorgan quantification rules:
    \begin{array}{rcl}<br />
 x \notin \left( {\bigcap\limits_{i = 1}^K {E_i } } \right) & \Leftrightarrow  & \left[ {\exists k,1 \le k \le K} \right]\left( {x \notin E_k } \right) \\ <br />
  & \Leftrightarrow  & \left[ {\exists k,1 \le k \le K} \right]\left( {x \in \left( {E_k } \right)'} \right) \\ <br />
  & \Leftrightarrow  & x \in \left( {\bigcup\limits_{i = 1}^K {\left( {E_i } \right)'} } \right) \\ <br />
 \end{array}.

    The other is by induction.
    \left( {E_1  \cap E_2 } \right)' = \left( {E_1 ' \cup E_2 '} \right).
    Then note \left( {\bigcap\limits_{i = 1}^{K + 1} {E_i } } \right) = \left( {\bigcap\limits_{i = 1}^K {E_i } } \right) \cap E_{K + 1} .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: October 19th 2010, 10:50 AM
  2. Replies: 0
    Last Post: June 29th 2010, 08:48 AM
  3. [SOLVED] direct proof and proof by contradiction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 27th 2010, 10:07 PM
  4. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: June 8th 2008, 01:20 PM
  5. proof that the proof that .999_ = 1 is not a proof (version)
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 14th 2008, 04:07 PM

Search Tags


/mathhelpforum @mathhelpforum