How do you determine whether a logical statement with nested quantifiers is true or not?

Question

I have the following statement

$$\neg\exists a\forall b \exists c (ab = b^2c + c)$$

where $a$,$b$,$c$ are all real numbers.

How do I determine whether its true or not, what is a good approach with these type of problems?

Answer 1

HINT: Standard rules for quantifiers give that your statement is equivalent to: $$ \forall a \exists b \forall c (ab\neq b^2+c) $$ Let us take $a=0$. Is it possible that exists $b$ such that for all $c$ we have $(b^2+1)c\neq0$?

Answer 2

I assume the negation affects everything inside, so we'll write this as $\neg[\exists a\forall b \exists c (ab = b^2c + c)]$

Let's be a little more tidy with this and write again as $\neg[\forall b, \exists a,c /(ab = b^2c + c)]$

So now we negate everything. For all becomes exists, exists becomes for all, and then you negate the equation. So that's

$$ \forall a,c, \exists b /(ab \neq b^2c + c)$$

Now this is something easier to prove either true or false (I'm betting on false)