Results 1 to 6 of 6

Math Help - Element in a Set

  1. #1
    Banned
    Joined
    Sep 2009
    Posts
    502

    Element in a Set

    Let A and B be nonempty sets, and let (x,y) \in A \times B.

    (x,y) \in A \times B \Rightarrow x \in A and y \in B is logically correct.


    Now if C and D are nonempty sets, and that (c,d) \notin C \times D.

    Will the following be true?

    (x,y) \notin C \times D \Rightarrow c \notin C and d \notin  D.
    Last edited by novice; February 22nd 2010 at 05:40 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by novice View Post
    Let A and B be nonempty sets, and let (x,y) \in A \times B.

    (x,y) \in A \times B \Rightarrow x \in A and y \in B is logically correct.


    Now if C and D are nonempty sets, and that (c,d) \notin C \times D.

    Will the following be true?

    (x,y) \notin C \times D \Rightarrow c \notin C and d \notin  D.
    (0,1)\notin\mathbb{N}^2 but 1\in\mathbb{N}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1584
    Awards
    1
    Quote Originally Posted by novice View Post
    Will the following be true?
    (x,y) \notin C \times D \Rightarrow c \notin C and d \notin  D.
    \begin{gathered}<br />
  \neg \left( {P \wedge Q} \right) \Leftrightarrow \left( {\neg P \vee \neg Q} \right) \hfill \\<br />
  \left( {a,b} \right) \notin A \times B \Leftrightarrow a \notin A \vee b \notin B \hfill \\ \end{gathered}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Sep 2009
    Posts
    502
    Quote Originally Posted by Plato View Post
    \begin{gathered}<br />
\neg \left( {P \wedge Q} \right) \Leftrightarrow \left( {\neg P \vee \neg Q} \right) \hfill \\<br />
\left( {a,b} \right) \notin A \times B \Leftrightarrow a \notin A \vee b \notin B \hfill \\ \end{gathered}
    Plato,
    Thank you for pointing the De Morgan's Law. While we are at it, I would like to discuss it with you in regard to a cross product.

    Since the De Morgan's Law said \sim (P \wedge Q) \equiv \sim P \vee \sim Q.

    Now let A and B be nonempty sets and let (x,y) \in \overline{A \times B}. We say that (x,y) \notin A \times B, which gives us x \notin A or y \notin B.

    So x\in \overline{A} and y \in \overline{B}. Consequently \overline{A} \times \overline{B}. We have just shown that \overline{A \times B} \subset \overline{A} \times \overline{B}, but according to the De Morgan's Law

    \overline{A \times B} \equiv \overline{A} \times \overline{B} is false.

    Suppose that I made \sim (P \wedge Q): \overline{A \times B}, it should follow that the logic equivalence is (\sim P \vee \sim Q): \overline{A} or \overline {B}.

    Consequently, \overline{A} or \overline {B} cannot be \overline{A} \times \overline{B}.

    Do you agree?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1584
    Awards
    1
    Quote Originally Posted by novice View Post
    Consequently, \overline{A} or \overline {B} cannot be \overline{A} \times \overline{B}.
    I think that last line is confusing.
    With appropriate domain restrictions, this is true: \overline A  \times \overline B  \subseteq \overline {A \times B} .
    But they need no be equal.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Sep 2009
    Posts
    502
    Quote Originally Posted by Plato View Post
    I think that last line is confusing.
    With appropriate domain restrictions, this is true: \overline A \times \overline B \subseteq \overline {A \times B} .
    But they need no be equal.
    With appropriate domain restrictions----Hmmm.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. element of N
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 12th 2012, 09:12 PM
  2. Orbits of an element and the Stabilizer of the element
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 24th 2011, 05:41 AM
  3. Least Element
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: November 21st 2010, 12:57 PM
  4. Replies: 3
    Last Post: March 23rd 2010, 07:05 PM
  5. element
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 9th 2009, 06:40 AM

Search Tags


/mathhelpforum @mathhelpforum