A group of 4 friends is designing a messaging application. Define a relation R on the set of these friends such that (x,y)∈R if and only if x can send a direct message to y through the app.
(i) If the app must allow messages to be sent only in pairs of friends who mutually agree (that is, whenever x can message y, y can message x), but sending a message to oneself is optional, identify the property satisfied by R and the property it need not satisfy.
(ii) Under the rules stated in (i), determine the total number of possible relations R on the set of 4 friends.
(iii) Suppose the designers further decide that no one should be able to message himself or herself. Keeping mutual messaging compulsory, calculate the number of possible relations under this new rule.