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Math Help - negation of statement

  1. #1
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    negation of statement

    Here is the statement about set of real numbers:
    (\forall x \in R) (x \neq 5 \rightarrow (\exists y \in R) (y < x))

    I need to find the negation of it.

    This is what I have so far. Am I correct?

    (\exists x \in R) (x = 5 \wedge (\forall y \exists R) (y > x))
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  2. #2
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    Quote Originally Posted by relyt View Post
    Here is the statement about set of real numbers:
    (\forall x \in R) (x \neq 5 \rightarrow (\exists y \in R) (y < x))

    I need to find the negation of it.

    This is what I have so far. Am I correct?

    (\exists x \in R) (x = 5 \wedge (\forall y \exists R) (y > x))
    i think the x \ne 5 should stay

    the negation of P \implies Q is P \wedge \sim Q

    so the antecedent does not change
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  3. #3
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    (\forall x \in R) (x \neq 5 \rightarrow (\exists y \in R) (y < x))

    So, would you say this like:
    "For the set of all real numbers x, if x does not equal 5 then their exists (at least) one y in the set of real numbers such that y is less than x."

    A is "For the set of all real numbers x, x not equal to 5."
    B is "There exists one y in the set of real numbers such that y is less than x"

    A \Rightarrow B \equiv B \vee \neg A.
    The negation is A  \wedge \neg B

     \neg B  \equiv  (\forall y \in R) (y > x)
    so,

     \equiv (\forall x \in R) (x \neq 5) \wedge (\forall y \in R) (y > x)

    And you would say, "For the set of all real numbers x, x not equal to 5 and the set of all real numbers y, y is greater than x."

    So, can you simplify this to, "for the set of all real numbers, y is greater than x, with the exception of x equals 5, where x is not defined"?
    Last edited by bmp05; March 29th 2009 at 04:25 AM.
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  4. #4
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    Quote Originally Posted by relyt View Post
    Here is the statement about set of real numbers:
    (\forall x \in R) (x \neq 5 \rightarrow (\exists y \in R) (y < x))
    I need to find the negation of it. This is what I have so far. Am I correct?
    (\exists x \in R) (x = 5 \wedge (\forall y {\color{red}\exists R}) (y {\color{red}>} x))
    Almost, two corrections.
    (\exists x \in R) (x \not= 5 \wedge (\forall y {\color{blue}\in R}) (y {\color{blue}\ge} x))
    Last edited by Plato; March 29th 2009 at 04:36 AM.
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  5. #5
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    A \Rightarrow B \equiv B \vee \neg A

    Are thes two statements equivalent?
    (\forall x \in \mathbb{R})(x \Rightarrow (\exists y \in \mathbb{R})(y)) \equiv (\exists y \in \mathbb{R})(y \vee (\exists x \in \mathbb{R})(x))
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  6. #6
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    Quote Originally Posted by bmp05 View Post
    A \Rightarrow B \equiv B \vee \neg A
    Are thes two statements equivalent?
    (\forall x \in \mathbb{R})(x \Rightarrow (\exists y \in \mathbb{R})(y)) \equiv (\exists y \in \mathbb{R})(y \vee (\exists x \in \mathbb{R})(x))
    I have no idea what those statements could possibly mean.
    But I do know that they are not equivalent.
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  7. #7
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    Quote Originally Posted by Plato View Post
    I have no idea what those statements could possibly mean.
    But I do know that they are not equivalent.

    Yes, I just found this link, which might be helpful: An Elementary Introduction to Logic and Set Theory: Methods of Proof

    And there's a section Rules of Replacement for Quantified Predicates with an example, which shows that what I wrote in my last post is definately wrong, but now, I wonder what the negation of (x) is?!
     \exists x \in \mathbb{R}: \neg (x) does that even make sense?
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  8. #8
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    Quote Originally Posted by bmp05 View Post

    I wonder what the negation of (x) is?!
     \exists x \in \mathbb{R}: \neg (x) does that even make sense?
     \neg (x) is just not proper notation. What does not x mean?
    Usually we have a propositional function.
    Example: Let P(x) mean that x is a rational number.
    Now  \neg P(x) reads “x is not a rational number.”
    \neg \left( {\forall x} \right)\left[ {P(x)} \right] reads: “It is false that every x is rational.
     \left( {\exists x} \right)\left[\neg {P(x)} \right] reads: “Some x is not rational.
    Clearly those two statements are equivalent.
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  9. #9
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    Yes, this is really quite interesting, so you can negate a predicate, but is there so much different between P(x):= (x \neq5) and P(y):=(x)..?

    So, the nagation of P  \forall x: x \neq 5 \equiv \exists x: x=5 \equiv x=5? But then the negation of Q  \forall x: x \equiv \neg Q  \equiv Because, you can negate all x, that would be for any x, False.



    Quote Originally Posted by Plato View Post
     \neg (x) is just not proper notation. What does not x mean?
    Usually we have a propositional function.
    Example: Let P(x) mean that x is a rational number.
    Now  \neg P(x) reads “x is not a rational number.”
    \neg \left( {\forall x} \right)\left[ {P(x)} \right] reads: “It is false that every x is rational.
     \left( {\exists x} \right)\left[\neg {P(x)} \right] reads: “Some x is not rational.
    Clearly those two statements are equivalent.
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