Q:

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.

