If she is my friend and you are her friend, then we are friends. Given this, the friend relationship in…

Friendship here is taken as mutual: if she is my friend, I am her friend, which is the commutative (symmetric) property. The given rule says that friends of a friend are friends, which is transitivity: if a R b and b R c then a R c. An equivalence relation also needs reflexivity, which the statement does not give, and 'implicative' is not a property of relations.