Condition (1) ⇒ A * B = A is the father of B. So X * Y = X is the father of Y. X is the father. Condition (2) ⇒ A # B = A is brother of B. So Y #Z = Y is Z’s brother. So X is also the father of Z. Condition (3) ⇒ A $ B = A is the son of B So Z $ W = Z is the son of W.