QUESTIQN 2: :1) Suppose that the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Express each of these sets with bit strings where the Em bit in the string is 1 if r' is in the set and 0 otherwise. (3 Marks) i) {3, 4, 5} a) {1, 3, 6, 10} iii) {2, 3, 4, 7, 8, 9} a) Suppose that the universal set is U: {1. 2, 3, 4, 5, 6. 7, 8, 9, 10}. nd the set Specied by each of these bit strings. (3 Marks) i)111100111] ii)010111 1000 iii) 10 0000 0001 e) A car dealer has 160 new cars. 50 cars have automatic transmission, 80 have the tilt steering and 30 have power windows. 42 have both automatic transmission and tilt steering. 18 of them have both automatic transmission and power windows. 15 cars have all 3 features. 65 cars have none of the above- mentioned features. Using Venn diagram nd the following: (8 Marks) i} How many have tilt steering and power windows only? a} How many have at least one of the features? Hi) How many have exactly 2 of the features? a) How many have exactly one of the feature? it) How many have power windows only? 'E a} Fuzzy sets are used in articial intelligence. Each element in the universal set U has a degree of membership, which is a real number between 0 and 1 (including 0 and 1), in a fuzzy set S. The fuzzy set S is denoted by listing the elements with their degrees of membership (elements with 0 degree of membership are not listed}- For instance, we write {06 Alice, 0.9 Brian, 0.4 Fred, 0.] Oscar, 0.5 Rita} for the set F (of famous people) to indicate that Alice has a 0.6 degree of membership in F, Brian has a 0.9 degree of membership in F, Fred has a 0.4 degree of membership in F, Oscar has a 0.1 degree of membership in F, and Rita has a 0.5 degree of membership in F (so that Brian is the most famous and Oscar is the least famous of these people). Also suppose that R is the set of rich people with R = {0.4 Alice, 0.8 Brian, 0.2 Fred, 0.9 Oscar, 0.? Rita}. Find the following a The complement of a fuzzyr set S is the set , with the degree of the membership of an element in S equal to 1 minus the degree of membership of this element in 5'. Find (the fuzzy set of people who are not famous) and (the fuzzy set of people who are not rich}. ii] The union of two fuzzy sets 5' and T is the fuzzy set S u T, where the degree of membership of an element in S u T is the maximum of the degrees of membership of this element in S and in T. Find the tzzy set F u R of rich or famous people. an The intersection of two fuzzy sets 5' and T is the fuzzy set S fl 1", where the degree of membership of an element in 3 Fl T is the minimum of the degrees of membership of this element in .5' and in it". Find the fuzzy set F F1 R of rich and famous people- (6 Marks} QUESTION 1: a) An explorer is captured by a group of cannibals. There are two types of cannibals-those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. He is allowed to ask the cannibal exactly one question. (5 Marks) 1 Explain why the question "Are you a liar?" does not work. in Find a question that the explorer can use to determine whether the cannibal always lies or always tells the truth. b) Let s, t and u denote the following simple statements: (4 Marks) s: Phyllis goes out for a walk. t: The moon is out. u: It is snowing. Translate each of the following into an English sentence. 1) (A -W) -5 ii) 1 - (-H -5) iii) -(5 - (1 v1)) is (-Iv-) Using laws of logic solve the following compound propositions. Also indicate the names of laws. (6 Marks) Using truth table verify whether [p-(4-)]-[-4)-(9-) is a tautology contradiction or contingency? (5 Marks) [Total Marks = 20] QUESTION 2: a) Suppose that the universal set is U = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10;. Express each of these sets with bit strings where the / bit in the string is 1 if i is in the set and 0 otherwise. (3 Marks) 1 {3, 4, 5} 1 {1, 3, 6, 10} 1) (2, 3, 4, 7, 8, 9} b) Suppose that the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10;. find the set specified by each of these bit strings. (3 Marks) 611 1100 1111 1 01 0111 1000 1) 10 0000 0001 car dealer has 160 new cars. 50 cars have