Are (∀x∈A)(∃y∈B)(x≤y) and (∃y∈B)(∀x∈A)(x≤y) the same? Do the statements $$(∀x∈A)(∃y∈B)(x≤y)$$ and $$(∃y∈B)(∀x∈A)(x≤y)$$ mean the same, even though the first two brackets are reversed?
P.S: Lets say, I have a sentence: There is no number from A, so it would be bigger than all numbers from B. 
 A: Nope they're not the same. In general, the meaning changes when permuting different quantifiers (however $(\forall x\in A)(\forall y\in B)$ and $(\forall y\in B)(\forall x\in A)$ are the same, and same holds for $\exists$).
To see why the two statements you gave have different meanings, start by translating them to English: the first one says "For every element of $A$ there is an element of $B$ larger than it" while the second says "There is an element of $B$ larger than every element of $A$". Notice that second implies the first, but not the first doesn't imply the second.
Now you should try to find an example of $A$ and $B$ subsets of, say, $\mathbb R$, such that the first statement holds but not the second.
A: These two statements do not mean the same thing. Let $A,B = \mathbb{N}$. 
The first statement is true: for any $x \in \mathbb{N}$ we can take $y=x+1$.
The second statement is false; there is no upper bound for the natural numbers. 
A: Example: let  $A=B= \mathbb R$ and $" \le "$ be the usual order on $ \mathbb R.$
If $x \in A$, then $y:=x+1 \in B$ and $x \le y.$
Hence we have  $(\forall x \in A)(\exists y \in B)(x \le y)$.
But  $(\exists y \in B)(\forall x \in A)(x \le y)$ means that $ \mathbb R$ is bounded from above.
A: $A=B=(0,1)$ is a counterexample. 
A: As Scientifica mentions in their answer, translating the logical statement to a sentence helps.
If you can simplify the sentence towards more natural language, do so.
Moreover, it is often instructive to to consider a special case.
And even better, combine the two and write sentences about a special case.
If $A=B=\mathbb N$, then the statements are:


*

*$(∀x∈A)(∃y∈B)(x≤y)$ — "for any natural number, there is a number that is at least as big"

*$(∃y∈B)(∀x∈A)(x≤y)$ — "there is a biggest natural number"


Written this way, the problem is pretty easy.
And that is often hard in math: rewriting a problem so that it becomes easy.
That can be hard!
This should also help you get a flavor of what might happen with more general sets $A$ and $B$.
Once you have a feel for the phenomenon from an example, more general cases are easier to make sense of.
If you have trouble figuring out why these are correct translations, let me know and I can give details.

You gave this sentence: "There is no number from A, so it would be bigger than all numbers from B."
This is literally $\neg(∃x∈A)(∀y∈B)(x>y)$.
Using the basic rules of negations and quantifiers, this can be seen to be equivalent with $(∀x∈A)(∃y∈B)(x≤y)$.
Indeed, only the first symbolic logical statement expresses the idea you have written in words.
If you end up with two candidates and are unsure like in this question, spelling them both out in natural language helps.
