Let the domain of discourse be all people on Facebook. You should use only the predicate Friend(x, y), which says that x and y are friends, and the predicates x = y and x $\neq$ y, which say whether or not x and y are the same persons. You may assume that friendship is a symmetric relationship, so Friend(x, y) and Friend(y, x) are the same. You can use constants to refer to specific people such as the “Bob” in the next example.
I am trying to translate the statement: "Two people have Bob as a mutual friend only if they are also friends"
I have come up with the following translation: ∀x(x $\neq$ Bob ^ Friend(x,Bob) ^ ∀y((y $\neq$ Bob)^ Friend(y, Bob) --> Friend(x,Bob) ^ Friend(y,Bob))). From the way I understand this, two people who are friends with Bob means that then they are mutual friends with Bob (statements after implication).
Is this the correct interpretation? Additionally, how would something like this be negated? I've tried utilizing DeMorgan's laws for Quatifiers but just got more and more confused.
Any help is appreciated!
I'd say (using just $F$ for friend and $b$ for Bob): $\forall x,y: \left( (x \neq y) \land F(x,b) \land F(y,b)\right) \to F(x,y)$
The first clause just says that $x,$ are distinct people having Bob as a friend, and then follows the necessary conclusion: they must already be friends of each other... (the "only if").
Your statement translation doesn't state "they are also friends", but reaffirms what you already know: that they're friends with Bob. So I'm pretty sure that's wrong. Also, the quantifiers are weirdly placed IMO. The way I'm doing it I state a "law" over the whole domain, which I think is intended.