How do the order of quantifiers affect the truth of a statement?

Question

I am just being introduced to quantifiers in logic and my lecturer was going through the following two statements. The question is to determine which, if any, is/are true.

1. $(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})[x + y = 0]$
2. $(\exists x \in \mathbb{R})(\forall y \in \mathbb{R})[x + y = 0]$

Clearly, the first statement is true; we can just let $y = -x$. However, my lecturer says that the second statement is false. I cannot wrap my head around why that is the case. If we can take $y = -x$ in the first, why can we not do the same for the second i.e. let $x = -y$? In fact, how is the second statement any different from the first?

Any intuitive explanations/examples would be greatly appreciated!

Answer 1

If you translate the sentences into English, you can see the differences between them.

The first sentence says "Every real number has an additive inverse." That is for every real number $x$, there is real number $y$ such that $x+y=0$. As you say, this is clearly true since we can take $y:=-x$.

By contrast, the second sentence says "There is a real number such that all real numbers are its additive inverse." That is, first we choose the value of $x$, and then every value of $y$ must then satisfy $x+y=0$. So your lecturer is right; once we specify the value of $x$ (which we have to do first since it is the first quantified variable in the expression), there is only one additive inverse of $x$ and so any other real number would not satisfy the equation.

Answer 2

The second one is false because there is no real number $x$ such that its addition with all real numbers results in $0$. You can take $y=x^2+1$ for each $x$ and observe that $$x+y=x^2+x+1\ne 0$$