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COMP232 - Mathematics for Computer Science Tutorial 8 Ali Moallemi moa [email protected] Iraj Hedayati h [email protected] Concordia University, Winter 2016 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 1/9 Table of Contents 1 2.5 Cardinality of sets Exercise 2 Exercise 6 Exercise 7 Exercise 8 Exercise 9 Exercise 10 Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 2/9 Exercise 2 Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the integers greater than 10 Answer: S = {11, 12, 13, . . . },S is countably infinite. f : Z+ → S, f (n) = n + 10. b) the odd negative integers Answer: S = {−1, −3, −5, . . . },S is countably infinite. f : Z+ → S, f (n) = −2n + 1. c) the integers with absolute value less than 1,000,000 Answer: S = {−999999, −999998, −999997, . . . , 999997, 999998, 999999}, S is finite. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 3/9 Exercise 2 Cont... Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. d) the real numbers between 0 and 2 Answer: S = (0, 2), S is uncountable. e) the set A × Z+ where A = {2, 3} Answer: S = {(2, 1), (3, 1), (2, 2), (3, 2), (2, 3), (3, 3) . . . },S is countably infinite. f : S → Z+ , f ((2, n)) = 2n and f ((3, n)) = 2n − 1. f) the integers that are multiples of 10 Answer: S = {. . . , −30, −20, −10, 0, 10, 20, 30, . . . },S is countably infinite. f : Z+ → S, f (2n) = (n − 1) · 10 and f (2n − 1) = −n · 10. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 4/9 Exercise 6 Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel. Answer: By closing even numbered rooms, we have: {1, 3, 5, . . .} All available room numbers are in form of 2 · k + 1. It is possible to move all the guest in odd room numbers to odd room numbers where k is an even number. Then the rest of odd room numbers where k is an odd number are free. Guests in even numbered rooms n move to 2 · (n − 1) + 1. Guests in odd numbered rooms n = 2k + 1 move to 2 · 2k + 1. New assignment of guests is as follows: Current room number k New room number 1=2·0+1 0 2·2·0+1=1 2 - 2·1+1=3 3=2·1+1 0 2·2·1+1=5 - 2·3+1=7 4 ... Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 5/9 Exercise 7 Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second building which also contains a countably infinite number of rooms. Show that the current guests can be spread out to fill every room of the two buildings of the hotel. Answer: Move the guest in room 2n to n, and Move the guest in room 2n + 1 to n in the new building. In simple words, keep guests in even numbered rooms in first floor. Move guests in odd numbered rooms to second floor. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 6/9 Exercise 8 Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest. Answer: Move the current guest in room n to 2n. Fill every odd room 2n + 1 with new guest number n. Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 7/9 Exercise 9 Suppose that a countably infinite number of buses, each containing a countably infinite number of guests, arrive at Hilbert’s fully occupied Grand Hotel. Show that all the arriving guests can be accommodated without evicting any current guest. Answer:Use the fact of having infinite number of prime numbers. Ref: http://mathandmultimedia.com/2014/05/26/grand-hotel-paradox/ Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 8/9 Exercise 10 Give an example of two uncountable sets A and B such that A − B is a) finite. Answer: A = R and B = R − {1}. Then A − B = {1}. b) countably infinite. Answer: A = R and B = R − Z. Then A − B = Z. c) uncountable. Answer: A = R and B = R− . Then A − B = R+ . Ali Moallemi, Iraj Hedayati COMP232 - Mathematics for Computer Science 9/9