if ∀x were false for all x , will it be true ?

for example:

∀x Q(x)

where Q(x) = ,,, x=x^2 and the domain consist of integers beside 0 and 1.

Question

if ∀x were false for all x , will it be true ?

for example:

∀x Q(x)

where Q(x) = ,,, x=x^2 and the domain consist of integers beside 0 and 1.

anonymous · Answer

also here's a question:

∃x(n=-n) and the domain consist of integers , find the truth value 

well it's false for every integer but what about 0 ? i know a 0 can't be negative but it will be the same as (n=n) ?

theeric · Answer

Hi! Math proofs is really cool, but it seems to take a special mode of thinking! An axiomatic way of thinking. I'm just okay at proving. But I might be able to help.

In your first question, I don't understand your notation. Sorry.
= ,,, x=
what does that mean?

With your second question, 
$n=-n=(-1)n$
So, consider $n=0$.
Let me prove this...
$n=0$
By multiplying by $-1$, we see that
$(-1)n=(-1)(0)$
$\implies-n=0$
Now $n=-n$ where $n=0$

Or you can just say
$n=0=(-1)0=(-1)n=-n$
$\implies n=-n$

theeric · Answer

For that second question, both ways are just as valid. I think.

anonymous · Answer

I mean that Q(x) is (x=x^2)

theeric · Answer

Oh! There's a name for that $x=x^2$ relationship! I forget what it is... I think it starts with an A, but I could be wrong.

anonymous · Answer

ah thank you man

theeric · Answer

So, here's the first question (restated)
For all $x$ where $x$ can be any integer except for $0$ and $1$, does $x=x^2$? If it does, then $x=x^2$ is true for that domain. If not, $x=x^2$ is false. And I do mean ALL $x$ must be true when we say "all $x$." Or $\forall x$, if you prefer.

theeric · Answer

For the second problem there, you also have to determine truth or falsity, but for each integer separately. I think you have the idea. Zero is the only number that makes it true,j and 0 is an integer, so it's in the domain.

I can prove that all other numbers won't work, but this problem doesn't ask for that.

theeric · Answer

Let me know if you have any questions.

anonymous · Answer

I think about the first question if it was all false then it's true , since all of x is the same

anonymous · Answer

@theEric  are you there ? :D

what do you think ? I think if " my first question " is false for all the domain and you see that 0 and 1 are not included so it's false for all x , and my proof said that if it's true for all x then it's true since we're talking about all x and if it's false for all x then it's also true.

theeric · Answer

The way I see it, is that it's a typical "TRUE or FALSE" statement type of question. Like the kind I usually saw in history class. Anyway. Now the statement is
For all $x$ in the domain (like $\infty ,...,-2, -1, 2, 3,..., \infty$) we see that $x=x^2$.
Do you see that for all $x$ in the domain?

anonymous · Answer

it's false for all x

theeric · Answer

It is false for all $x$!

theeric · Answer

$x=x^2$ is false for all $x$!

theeric · Answer

So, the statement is false in it's entirety.

anonymous · Answer

and this is my question , if we're asked to give the truth value of ∀x for this one , will it be true or false ?

theeric · Answer

I say false.
It can be true ONLY IF ALL $x$ in the domain make it true. Turns out no $x$ makes it true. So, the statement is false.

anonymous · Answer

thank you ^^

theeric · Answer

You're welcome!

theeric · Answer

Good luck with any other problems that you have to do!