Is the Logical Statement \forall x[x^2\leq 0\Longrightarrow x>0] True or False?

[evagelos]

Is the following statement true or false??

$$\forall x[x^2\leq 0\Longrightarrow x>0]$$

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


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

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

So there exists an element x=0.Hence the negation is true and thus the above statement is false

 

[mathman]

Solution no 1 is incorrect because x²=0 does not imply x >0. I.e. the statement is not vacuously satisfied.

 

[evagelos]

Solution no 1 is incorrect because x²=0 does not imply x >0. I.e. the statement is not vacuously satisfied.

Why, is it not $$x^2\leq 0$$always false and hence whether x>0 is true or false $$x^2\leq 0\Longrightarrow x>0$$ is always true??

 

[Matthollyw00d]

Why, is it not $$x^2\leq 0$$always false and hence whether x>0 is true or false $$x^2\leq 0\Longrightarrow x>0$$ is always true??

It is not always false though, ie when x²=0.

 

[matheinste]

Is the following statement true or false??

$$\forall x[x^2\leq 0\Longrightarrow x>0]$$

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.

Matheinste.

 

[CRGreathouse]

Is the following statement true or false??

$$\forall x[x^2\leq 0\Longrightarrow x>0]$$

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.