How to read sentences with three quantifiers and check for their truth?

Question

I am trying to solve an exercise in which I must decide whether the following sentences are true:

• $\forall b \; \exists a \; \forall x \; (x^2 + ax + b > 0)$

• $\exists b \; \forall a \; \exists x \; (x^2 + ax + b = 0)$

• $\exists a \; \forall b \; \exists x \; (x^2 + ax + b = 0)$

But I don't know how to parse them to check if they are indeed true. I guess I know how to parse two quantifiers easily but this is kinda confusing to me.

Answer 1

For the first one, it reads as follows:

For any $b$, there exist $a$ such that for any $x$, $$x^2 + ax + b > 0.$$

Perhaps by graphing this you can see this statement is most certainly false for $b \le 0$. However, to show the statement is false, we can show that its negation is true.

Negation: There exists $b$ such that for any $a$, there exist an $x$ such that $$x^2 + ax + b \le 0.$$

So, it suffices to show that the negation is true. Take $b=0$ and for any $a$ choose $x=0$. Then $x^2 + ax = x(x+a) = 0 \le 0$.

Hopefully you can see now how to do the other two.

Answer 2

I often find it helpful to mentally insert "such that" after $\exists$.

The first example you gave I would read as "for all $a$, there exists a $b$ such that for all $x$ ...."

Another way to think you this, is that I am giving you an $a$, and you have to find a $b$ such that if I then give you any $x$, the rest of the statement will hold.

Example 2 would read "there exists a $b$ such that for every $a$, there exists an $x$ such that..."

And example three would be the same, but with $a$ and $b$ swapped around.

Let me know if this helps!