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Really a Question about Notation

  1. Apr 7, 2014 #1

    Atomised

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    1. The problem statement, all variables and given/known data

    Form negation and then either prove statement or negation:

    [itex]\forall[/itex]y [itex]\in[/itex] {[itex] x | x \in Z, x>=1[/itex]}, [itex]5y^2+5y+1[/itex] is a prime number.

    3. The attempt at a solution

    Answer given:

    [itex]\exists[/itex]y [itex]\in[/itex] {[itex] x | x \in Z, x>=1[/itex]} such that [itex]5y^2+5y+1[/itex] is not prime.

    The negation is true, counterexample follows.

    My question is why are [itex]x,y[/itex] mixed up?
     
  2. jcsd
  3. Apr 7, 2014 #2
    What do you mean why are [itex]x, y[/itex] mixed up?
     
  4. Apr 7, 2014 #3

    Atomised

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    Well, there exists y belonging to a set where the elements are defined by x...

    Would it be valid to express it as there exists x belonging to a set where the elements are defined by x... or the above the preferred way of doing it?
     
  5. Apr 7, 2014 #4
    The way I interpret it is like this:

    Let [itex]X[/itex] be defined as the set of numbers [itex]\{x|x\in\mathbb{Z}, x\geq1\}[/itex].

    Then:

    [itex]\exists y \in X \text{ such that } 5y^2+5y+1 \text{ is not prime}[/itex]

    This might make it clearer that the set is being defined by the dummy variable [itex]x[/itex]. It has no real purpose besides just being a dummy variable from what I can see.
     
  6. Apr 7, 2014 #5

    Atomised

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    Very helpful thank you.
     
  7. Apr 7, 2014 #6

    Ray Vickson

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    In plain English, the statement is: ##5 y^2 + 5y+1## is prime for any positive integer ##y##. Of course, as you have shown, the statement is false.
     
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