Results 1 to 7 of 7

Math Help - The negation of infinite

  1. #1
    Banned
    Joined
    Sep 2009
    Posts
    502

    The negation of infinite

    What is the correct negation of the statement

    P: There are infinitely many integers n such that \sqrt{n} is irrational.

    Is it correct to say \sim P: There exist no integers n such that \sqrt{n} is irrational.

    or

    Is it correct to say \sim P: There is a finite set of integers n such that \sqrt{n} is irrational.
    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
    What is the correct negation of the statement

    P: There are infinitely many integers n such that \sqrt{n} is irrational.

    Is it correct to say \sim P: There exist no integers n such that \sqrt{n} is irrational.

    or

    Is it correct to say \sim P: There is a finite set of integers n such that \sqrt{n} is irrational.
    The second.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1
    I disagree. Nether of those is the negation.
    It is: \left( {\exists j \in \mathbb{Z}^ +  } \right)\left( {\forall n \in \mathbb{Z}^ +  ,n > J \Rightarrow \sqrt n  \in \mathbb{Q}} \right).
    That is, there is at most a finite set of integers each of which has an irrational square root.
    Or: “the set of integers having irrational square roots is finite."

    The statement “There is a finite set of integers each having a irrational square root” is always true.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Sep 2009
    Posts
    502
    Quote Originally Posted by Plato View Post
    I disagree. Nether of those is the negation.
    It is: \left( {\exists j \in \mathbb{Z}^ + } \right)\left( {\forall n \in \mathbb{Z}^ + ,n > J \Rightarrow \sqrt n \in \mathbb{\overline Q}} \right).
    That is, there is at most a finite set of integers each of which has an irrational square root.
    Or: “the set of integers having irrational square roots is finite."

    The statement “There is a finite set of integers each having a irrational square root” is always true.
    Plato,
    I put a bar over your \mathbb{Q} . I am just guessing because someone told me that there is no symbol for irrational numbers.
    Be that as it may, what is n > J?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,649
    Thanks
    1597
    Awards
    1
    Quote Originally Posted by novice View Post
    Plato,
    I put a bar over your \mathbb{Q} . I am just guessing because someone told me that there is no symbol for irrational numbers.
    Be that as it may, what is n > J?
    Why add an overline?
    That says that every integer greater than J has a rational square root.
    That means there are only finitely many integers having irrational square roots.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    I'm sorry. I guess I didn't read the actual responses close enough.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Banned
    Joined
    Sep 2009
    Posts
    502
    Quote Originally Posted by Drexel28 View Post
    I'm sorry. I guess I didn't read the actual responses close enough.
    It's perfectly alright. I am very grateful that you always help me.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Negation
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: October 20th 2010, 12:10 PM
  2. Negation of x in A or B
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 13th 2010, 08:13 AM
  3. Negation help (2)
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: April 15th 2010, 07:49 AM
  4. Negation help
    Posted in the Discrete Math Forum
    Replies: 14
    Last Post: April 13th 2010, 09:20 AM
  5. The Negation
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 22nd 2009, 12:55 PM

Search Tags


/mathhelpforum @mathhelpforum