Question 15 (2 points) Fuzzy sets are used in artificial 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 {0.6 Alice, 0.9 Brian, 0.4 Fred, 0.1 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 = 10.4 Alice, 0.8 Brian, 0.2 Fred, 0.9 Oscar, 0.7 Rita}. The intersection of two fuzzy sets S and T is the fuzzy set S n T, where the degree of membership of an element in S n T is the minimum of the degrees of this element in S and in T. Which of the following fuzzy sets represent Fn R. ONone of the other choices 10.4 Alice, 0.8 Brian, 0.2 Fred, 0.1 Oscar, 0.5 Rita} 10.6 Alice, 0.9 Brian, 0.4 Fred, 0.9 Oscar, 0.7 Rita} 10.5 Alice, 0.85 Brian, 0.3 Fred, 0.5 Oscar, 0.6 Rita} 10.2 Alice, 0.1 Brian, 0.2 Fred, 0.8 Oscar, 0.2 Rita} {1 Alice, 1.7 Brian, 0.6 Fred, 1 Oscar, 1.2 Rita} {Alice, Brian, Fred, Oscar, Rita}Question 16 (2 points) Fuzzy sets are used in artificial 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 {0.6 Alice, 0.9 Brian, 0.4 Fred, 0.1 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 = 10.4 Alice, 0.8 Brian, 0.2 Fred, 0.9 Oscar, 0.7 Rita}. The union of two fuzzy sets S and T is the fuzzy set S U T, where the degree of membership of an element in S U Z is the maximum of the degrees of this element in S and in T. Which of the following fuzzy sets represent FUR. 10.6 Alice, 0.9 Brian, 0.4 Fred, 0.9 Oscar, 0.7 Rita} 10.2 Alice, 0.1 Brian, 0.2 Fred, 0.8 Oscar, 0.2 Rita} 10.5 Alice, 0.85 Brian, 0.3 Fred, 0.5 Oscar, 0.6 Rita} Of1 Alice, 1.7 Brian, 0.6 Fred, 1 Oscar, 1.2 Rita} None of the other choices 10.4 Alice, 0.8 Brian, 0.2 Fred, 0.1 Oscar, 0.5 Rita} {Alice, Brian, Fred, Oscar, Rita}Question 17 (2 points) Determine which of the following choice provides an example of disjoint sets. '17:} {1,2,3,4,5,6,7,8,9] and {1,3,5} {a,b,c,d,e} and {f,g,h} '17:} {1,2345} and {0,540} {a,c,e,g,i,k} and {b,c,d,e} Question 18 (2 points) Determine which of the following choice provides an example of disjoint sets. {a,b,c,d} and {e,f,g} O [10,12,14,16,18,20} and [0,2,4,6,8,10} {1:} {1,234.5} and [0,2,4,6,8,10} (:3. {1,2,3,4,5,6,7,8} and {2,4,6} Question 19 (2 points) Which of the following sets represents the union of the sets {1,3,5,7,9} and {2,4,6,8}. {1,3,5, 7,9} 12,4,6,8} {1,2,3,4,5,6,7,8,9} ONone of the above Question 20 (2 points) Let A = {3,4,5,6,7,8}, B = {2,4,6,8,10}, and U={1,2,3,4,5,6,7,8,9,10} where U is the universal set. Determine which of the following numbers are elements of A D B. 0 1 2 3 0 4 75 07 10Question 21 (2 points) Determine which of the following elements would be in the union of the sets {1.2,a.b,c} and {a,f,g,3,4,6,8} 1 N 2 Question 22 (2 points) Determine which of the following elements would be in the intersecion of the sets {1,2,3,4,5] and {4,5,6,7,8} \\OOONONU'I-P-EONH Question 23 (2 points) Let A = {1,2,3}, B = {a,b}, and C = {x,y). Select the objects below that represent elements of A X (B X C). O (1,a,x) O ((b, x), 1) (3, (b, x)) O (1,a,y) 2, (a,x) Of1,b,x} O {1,{b,x}} ((2,a), y) O(13), (a,x)) O ({1},a,x)Question 24 (2 points) Below is given a proof of a result. Proof: Assume that x is even. Then x = 23 for some integer 3. So 33.3 43:: 5 = 3(2a)2 4(2a.) 5 2 12a2 8a 5 = 2(6a2 Since 60.2 4a 3 is an integer, 3:32 4m 5 is odd. Determine which of the following statementls) were proven by the above proof. l:l If 33:2 4:13 5 is even, then x is even. Cl If 3:1:2 4m 5 is odd, then x is odd. l:l If x is even, then 3m2 4m 5 is odd. l:l If x is odd, then 33:2 41$ 5 is even. l:l If x is even, then 3$2 4m 5 is even. l:l If 33:2 435: 5 is even, then x is odd. Cl lfx is odd, then 3m2 4m 5 is odd. l:l If 335:2 4:15: 5 is odd, then x is even. Question 25 (2 points) Below is given a proof of a result. Proof: Assume that x is odd. Then x = 2b+1 for some integer b. 50 33:2 4:13 5 = 3(2b+ 1)2 4(2b+ 1) 5 = 3(452 +4b+1) Sinceb2 -| 2b 3 is an integer, 3:32 4m 5 is even. Determine which of the following statementls) were proven by the above proof. ._/ If 3:132 4a: 5 is even, then x is even. ._, If x is even, then 3332 4m 5 is odd. .i, \"3132 4:1: 5 is odd, then x is even. ._. If 3m2 4a: 5 is odd, then x is odd. ._, lfx is odd,then 3:122 4x 5 is odd. .7, If 3132 4:1: 5 is even, then x is odd. .7, If x is odd, then 3:132 433 5 is even. ._, If x is even, then 3232 4m 5 is even. Question 27 (2 points) When attempting to prove the statement "Ifx and yare real numbers such thatm -l- y 2 2, then x 2 1 or y 2 1." by contraposition, determine which of the following options describes such an approach. '1; We assume that x and y are real numbers such that m -l- y 2 2, and we attempt to show that m 2 1 mg 2 1 by using axioms, definitions, previously proven theorems, and rules of inference. II; We assume that x and y are real numbers such that (I: 2 1 or y 2 1, and we attempt to show that m -|- y 2 2 by using axioms, definitions, previously proven theorems, and rules of inference. 'C; We attempt to show the existence of real numbers x and y such that m+y22,wherem210r'y21. It; We assume that x and y are real numbers such that :1\": + y 2 2, and we attempt to show that x