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What is the negation of the statement

  1. Dec 20, 2003 #1
    What is the negation of the statement "For each s in R, there exists an r in R such that if f(r) >0, then g(s) >0."

    The answer is "There exists an s in R such that for each r in R, f(r) >0 and g(s) <0."

    What is the general method to find the negation of any logical statement?

    Thanks!!
     
  2. jcsd
  3. Dec 20, 2003 #2

    Doc Al

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    Staff: Mentor

    While I can't give you a general method, you may find it useful to review the concept of contradictory statements from Boolean logic:

    All S is P is contradictory to Some S is not P

    No S is P is contradictory to Some S is P

    A statement and its contradictory cannot both be true (or both be false). Thus if "All S is P" is not true, then "Some S is not P" must be true. Of course, this only applies to statements that can be put in standard categorical form.
     
  4. Dec 20, 2003 #3

    Hurkyl

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    Staff Emeritus
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    Gold Member

    Basically, you just want to distribute the negation. Use the laws

    [tex]\neg \forall x: P(x) = \exists x: \neg P(x)[/tex]
    [tex]\neg \exists x: P(x) = \forall x: \neg P(x)[/tex]
    [tex]\neg(x \wedge y) = \neg x \vee \neg y[/tex]
    [tex]\neg(x \vee y) = \neg x \wedge \neg y[/tex]
    [tex]\neg(x \Rightarrow y) = x \wedge \neg y[/tex]
    [tex]\neg(\neg x) = x[/tex]
     
    Last edited: Dec 20, 2003
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