5 Welcome to your new job at Hotel Gemini, where we aim to be Orlando's most popular hotel and resort accommodation. As you know, we have 1000 hotel rooms and they are numbered 1,2,,1000. Next week is our grand opening. To celebrate, the Florida Logic and Proof Committee will be our first guests. The committee has M members: member 1 , member 2 , and so on up to member M. We may sometimes refer to member m as guest m. The committee has some very important members, and so there may be nights when one member will stay in more than one room. Also, to save money, some committee members will need to share rooms on some nights. (a) On Monday, the first 20 members of the committee will arrive. They request that member 1 stay in room 1 and that member m{2,3,,20} stay in room i anytime i is a prime factor of m. Recall that the set of prime numbers is {2,3,5,}. Write down a relation R that describes the incoming guest room assignments. Every ordered pair in R should have the form (member, room number). Identify the domain and range of R. (b) On Tuesday, five more members will arrive and some returning guests will be putting in for room changes. The committee requests that the new room assignments be as follows: Member m{1,2,3,,25} will stay in room i anytime i is congruent to m modulo 100. Write down a relation in double set union form R=k=j=(,) that describes the guest room assignments. How many rooms will each guest be staying in? c) On Wednesday, five more members will arrive and again some returning guests will be putting in for room changes. Now, the committee requests that member m{1,2,3,,30} stay in room i anytime i is the highest power of 2 that divides m (see Problem 2). Write down a relation R that describes the guest room assignments. Is R a function? d) On Thursday, the first 500 members of the committee will be here. They are requesting that member m{1,2,3,,500} stay in room m+250 if m250 and in room 1000m if m>250. Write down a relation R that describes the guest room assignments. Is R a bijective function between the set of guests and the set of occupied hotel rooms? e) On Friday, each of the 800 committee members that will be here will want exactly one room and nobody wants to share a room. How many ways can we make room assignments for their stay that night? f) On Saturday, the full delegation of M committee members will be here. How many ways can we make room assignments for their stay that night? Of course, every member must stay in at least one room, and we continue to allow for the possibility that members can book more than one room and that they can also share rooms. 5 Welcome to your new job at Hotel Gemini, where we aim to be Orlando's most popular hotel and resort accommodation. As you know, we have 1000 hotel rooms and they are numbered 1,2,,1000. Next week is our grand opening. To celebrate, the Florida Logic and Proof Committee will be our first guests. The committee has M members: member 1 , member 2 , and so on up to member M. We may sometimes refer to member m as guest m. The committee has some very important members, and so there may be nights when one member will stay in more than one room. Also, to save money, some committee members will need to share rooms on some nights. (a) On Monday, the first 20 members of the committee will arrive. They request that member 1 stay in room 1 and that member m{2,3,,20} stay in room i anytime i is a prime factor of m. Recall that the set of prime numbers is {2,3,5,}. Write down a relation R that describes the incoming guest room assignments. Every ordered pair in R should have the form (member, room number). Identify the domain and range of R. (b) On Tuesday, five more members will arrive and some returning guests will be putting in for room changes. The committee requests that the new room assignments be as follows: Member m{1,2,3,,25} will stay in room i anytime i is congruent to m modulo 100. Write down a relation in double set union form R=k=j=(,) that describes the guest room assignments. How many rooms will each guest be staying in? c) On Wednesday, five more members will arrive and again some returning guests will be putting in for room changes. Now, the committee requests that member m{1,2,3,,30} stay in room i anytime i is the highest power of 2 that divides m (see Problem 2). Write down a relation R that describes the guest room assignments. Is R a function? d) On Thursday, the first 500 members of the committee will be here. They are requesting that member m{1,2,3,,500} stay in room m+250 if m250 and in room 1000m if m>250. Write down a relation R that describes the guest room assignments. Is R a bijective function between the set of guests and the set of occupied hotel rooms? e) On Friday, each of the 800 committee members that will be here will want exactly one room and nobody wants to share a room. How many ways can we make room assignments for their stay that night? f) On Saturday, the full delegation of M committee members will be here. How many ways can we make room assignments for their stay that night? Of course, every member must stay in at least one room, and we continue to allow for the possibility that members can book more than one room and that they can also share rooms