Q:

Difference Between "$\forall x \exists y$" and "$\exists y \forall x$"

Possible Duplicate:
Confused between Nested Quantifiers 

I asked the question about two sentences.
 interpreting mixed quantifier
But, I don't know the meaning difference between
$$∀x∃y(\text{Cube}(x) → (\text{Tet}(y) ∧ \text{LeftOf}(x, y))),$$ and
$$∃y∀x(\text{Cube}(x) → (\text{Tet}(y) ∧ \text{LeftOf}(x, y))),$$
"Every cube is to the left of a tetrahedron"
"There is a tetrahedron that is to the right of every cube"
I think these sentences have same meaning.
Is it wrong? please give me your opinion.

A:

Let’s look at a simpler example, where we assume that the variables range over real numbers: $\forall y\exists x(x+y=0)$ and $\exists x\forall y(x+y=0)$. The first says that every real number has an additive inverse, which is true. The second says that there is some particular real number $-$ call it $z$, say $-$ such that $z+y=0$ no matter what $y$ is; that’s clearly false.
In general you cannot reverse $\forall$ and $\exists$.

A:

The two sentences in your question do not have the same meaning: consider an infinite sequence
$$\text{Cube}\qquad\text{Tetrahedron}\qquad\text{Cube}\qquad\text{Tetrahedron}\qquad\cdots$$
Every cube is to the left of a tetrahedron, but there is no tetrahedron that is to the right of every cube.

