I'm working through "How to prove it" by Daniel Velleman. We're asked to translate the following sentence into it's logical equivalent. Anyone who has a friend who has measles will have to be quarantined.
My answer would be: Let
- $F(y,x)$ mean $y$ is a friend of $x$
- $M(y)$ mean $y$ has measles
- $Q(x)$ mean $x$ must be quarantined.
Then the statement is $ \forall x \forall y [(F(x,y) \land M(y)) \implies Q(x)] $
The book gives the answer: $ \forall x [\exists y(F(y,x) \land M(y)) \implies Q(x)] $
To me the book's answer reads as: There exists someone who has a friend who has measles. This seems different to the original statement. Could someone help me understand the provided answer and how my answer is wrong?