SOLVE IT

1.1: Which of the following is always true for any sets A and B ? a) AB=AB b) AB=AB c) AB=AB d) AB=BA e) None of the above 1.2: The inverse of function f(x)=x5+2 is a) f1(x)=(x+2)1/5 b) f1(x)=(x2)1/5 c) f1(x)=x5 d) f1(x)=(x52) e) None of the above 1.3: The relation R={(x,y)x=y} on the set of integers is a) Reflexive only b) Symmetric only d) Equivalence relation c) Transitive only e) None of the above 1.4: The number of vertices in the complete bipartite graph K4,5 is a) 9 d) 40 b) 10 e) None of the above c) 20 1.5: The number of edges of a full binary tree with 13 internal vertices is a) 12 d) 27 b) 14 e) None of the above c) 26 Each question is worth 10 marks. Answer the following questions: Q-2: is a tautology, contradiction or contingency: i. (ab)(ab)(ab), by using truth tables. ii. (p(qp)), by using laws of logic. b) [2 marks] Determine whether the following quantifier is True or False, Justify your answer: m ( m is prime 2m), domain is the set of integers. Q-3: set of integers defined by f(x)=x+3 and g(x)=3x4 MT131/M131-Final Page 2 of 3 2022/2023 Fall i. Show that f is one-to-one and find its inverse. ii. Find the composition of f and g. b) [22 marks ] Let a=2340 and b=1188. Find lcm(a,b) and gcd(a,b) Q-4: a) [23 marks] Let R={(1,1),(1,2),(1,4),(2,1),(2,3),(3,2),(3,4), (4,1),(4,3),(4,4)} and S={(1,1),(1,3),(2,2),(3,3),(3,4),(4,1), (4,4)} be relations on A={1,2,3,4}. Find the matrix representation b) [4 marks] Let R be a partial order relation on A={1,2,3,4,5,6},(3,2),(3,4) where aRb means ab. Draw the Hasse diagram for the relation R. (3,1), Q-5: a) [5 marks] Find the adjacency matrix of the given graph with vertices 1,2,3,4,5,6: 2253321312233111 b) [ 51 marks] Use the following tree to answer the questions below: i. What is the height of the tree? ii. What is the level of N? iii. How many siblings does B have? iv. How many descendants does B have? v. Is the tree balanced