What does this logic evaluate to in plain English?

$$\forall x \exists y P(x, y)$$

where the universe is the students in a class and P(x, y) means student x copies off of y.

Does this mean all students in the class each have their own person that they copy off of? Or does it mean all students in the class together copy off of a single, specific smart person?

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It means that given any student, there is someone they copy from. It doesn't have to be "their own" (that suggests that different students copy off different people, and this is not a given). To write that there is a single person who everyone copies from you would write $$\exists y \forall x P(x,y)$$ (there is a $y$ such that for all $x$, $x$ copies off $y$). This is not precluded from being the case by $\forall x\exists y P(x,y)$, but it is not required that this be the case in order for $\forall x \exists y P(x,y)$ to be true. – Arturo Magidin Apr 28 '12 at 4:11

Maybe it will help if you read $\forall x\exists yP(x,y)$ as $\forall x \exists y_xP(x,y_x)$, meaning that for every $x$, there is a $y$, which in general depends on $x$, such that $x$ copies from $y$. For a given $x$, there may be several such $y$, like all the near neighbours during test-taking.
The assertion that there is someone that everyone copies from would be written $\exists y\forall x P(x,y)$.