Let F(x,y) be the statement "x can fool y," where the domain consists of all people in the world.
∃y1∃y2 (F(Nancy,y1) ∧ F(Nancy,y2) ∧ y1 ≠ y2 ∧ ∀y(F(Nancy,y) → (y=y1 ∨ y=y2)))
Why this proposition denotes: Nancy can fool exactly two people.
I do not understand the way that we are using with quantifier to denote number of people.
To express the same meaning, can I say this instead:
∃y1∃y2 (F(Nancy,y1) ∧ F(Nancy,y2) ∧ y1 ≠ y2)
The second proposition is not equivalent, because it does not limit the number of people Nancy can fool. There may exist a $y_3$ not equal to $y_1$ or $y_2$ such that $F(\textrm{Nancy},y_3)$ is true.
The statement $\forall y : F(\textrm{Nancy},y) \Rightarrow (y = y_1 \lor y = y_2)$ means that if $F(\textrm{Nancy},y)$ holds, either $y$ is $y_1$ or $y$ is $y_2$. In other words, if Nancy can fool person $y$, either $y$ is person $y_1$ or person $y_2$. Or, equivalently, if person $y$ is not $y_1$ or $y_2$, Nancy cannot fool $y$. The rest of the proposition establishes that $y_1$ and $y_2$ can both be fooled by Nancy, and $y_1$ and $y_2$ are two different people. Hence, Nancy can fool exactly two people.