1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Logical Quantifiers: For all such that

  1. Jan 21, 2013 #1
    1. The problem statement, all variables and given/known data

    Negate the following statement:
    For all x such that 0 < x < 1, f(x) <2 or f(x) > 5.


    2. Relevant equations

    I understand the universal quantifier is used as, "For all x, p(x)."
    and the existential quantifier is used as, "There exists x such that p(x)."

    I understand how to negate these alone; however, in this problem I am confused by "For all x such that..."


    3. The attempt at a solution

    Symbolic Statement:
    \forall x \ni 0 < x < 1, f(x) < 2 or f(x) > 5.



    Negation 1:

    \exists x \ni 0 < x < 1, [f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5].


    Negation 2:

    \exists x \forall [x = 0 or x < 0] and [x = 1 or x > 1], [f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5].
     
  2. jcsd
  3. Jan 21, 2013 #2
    So, how would you negate "For all x, p(x)."?
     
  4. Jan 21, 2013 #3
    Statement: For all x, p(x).

    Negation: There exists an x such that ~p(x).
     
  5. Jan 21, 2013 #4

    pasmith

    User Avatar
    Homework Helper

    In strict technicality, you don't introduce an object without saying what it is. Thus you should never have "for all x" but "for all x in X". The negation of "for all x in X, P(x)" is "there exists x in X such that not P(x)".

    In your example, assuming x is supposed to be real, [itex]x \in X = \{y \in \mathbb{R}: 0 < y < 1\}[/itex] which in interval notation is [itex]x \in (0,1)[/itex].

    You need "x \in (0,1)" instead of "\ni 0 < x < 1", but otherwise this is correct. You can however simplify "[f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5]" to "[itex]2 \leq f(x) \leq 5[/itex]".
     
  6. Jan 21, 2013 #5
    Thanks pasmith!

    That cleared up a lot for me. Much appreciated!
     
  7. Jan 22, 2013 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    To answer your specific question, the negation of "for all x, P(x)" is "there exist x such that NOT P(x)"
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Logical Quantifiers: For all such that
Loading...