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True,false statement

  1. Jun 17, 2010 #1
    Is the following statement true or false??

    [tex]\forall x[x^2\leq 0\Longrightarrow x>0][/tex]

    Solution No 1: since [tex]x^2\geq 0[/tex] for all ,x the above statement is "vacuously satisfied"

    Solution No 2: the negation of the above statement is:

    [tex]\exists x[x^2\leq 0[/tex] and [tex]x\leq 0[/tex] .But since [tex]x\leq 0\Longrightarrow x^2\geq 0[/tex].So we have : [tex]x^2\geq 0[/tex] and [tex]x^2\leq 0[/tex] ,which implies that x=0.

    So there exists an element x=0.Hence the negation is true and thus the above statement is false
    Last edited: Jun 17, 2010
  2. jcsd
  3. Jun 18, 2010 #2


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    Solution no 1 is incorrect because x2=0 does not imply x >0. I.e. the statement is not vacuously satisfied.
  4. Jun 18, 2010 #3
    Why, is it not [tex]x^2\leq 0[/tex]always false and hence whether x>0 is true or false [tex]x^2\leq 0\Longrightarrow x>0[/tex] is always true??
  5. Jun 19, 2010 #4
    It is not always false though, ie when x2=0.
  6. Jun 19, 2010 #5
    I am fairly new to logical terminology.

    As written I would say yes.Yes. It is true or false.

    As spoken it depends on the vocal delivery how the question is interpreted.

    What are the x.

    Do you mean for all real numbers assign a value of true or false to it.

    What sort of implication are you using.

    Normally, assuming natural numbers are meant, as per truth table I would assign a truth value of 1 to it. A false antecedent always implies a truth value of 1 to the implication. Of course in the normal course of language most of us look for a causal relation between the antecedent and the consequent (maybe wrong terminology) in which case the outcome is different.

  7. Jun 20, 2010 #6


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    Assuming your universe of discourse is the real numbers (or, for that matter, the integers), the statement is false since there is an element such that x^2 <= 0 but not x > 0.
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