BIT 1206 BAC 1201 BISF 1101 BSD 1202 DISCRETE MATHEMATICS.
UNIVERSITY EXAMINATIONS: 2021/2022 ORDINARY EXAMINATION FOR BACHELOR OFSCIENCE INFORMATION TECHNOLOGY, BSc. APPLIED COMPUTING, BSD, BISF BIT 1206/ BAC 1201/ BISF 1101/ BSD 1202: DISCRETE MATHEMATICS (PARTTIME/ DISTANCE LEARNING) DATE: DECEMBER, 2021 TIME: 2 HOURS INSTRUCTIONS: Answer Question One and Any Other Two questions QUESTION ONE (a)Use the laws of algebra of sets to simplify [5Marks] (b)Prove : if 𝐴 is a subset of the null set 𝜙 ,then 𝐴 = 𝜙. [5Marks] (c)Let 𝑆 be a subset of 𝐴 and let ⅈ: 𝑆 ↦ 𝐴 be an iclusion map .Show that the inclusion map is one-to-one. [5Marks] (d)Find the largest interval 𝐷 on which the formulae 𝑓(𝑥) = 𝑥 2 defines a one to one function. [5Marks] QUESTION TWO (a)Find a traversable trail 𝛼 for the graph 𝐺 where𝑉(𝐺) = {𝐴, 𝐵, 𝐶,𝐷} and 𝐸(𝐺) = [{𝐴, 𝐶},{𝐴,𝐷},{𝐵, 𝐶},{𝐵,𝐷},{𝐶,𝐷}] . [5Marks] (b)Draw the multigraph 𝐺 whose adjacency matrix is [10Marks] QUESTION THREE (a)Determine the number of loops and multiple edges in a multigraph 𝐺 from its adjacency matrix [5Marks] (b)Let 𝐺 be a graph with 𝑚 vertices. Describe two major drawbacks in the computer storage of 𝐺 as its adjacency matrix 𝐴. [10Marks] QUESTION FOUR (a)Prove:There is a path from a vertex 𝑢 to a vertex 𝑣 if and only if there is a simple path from 𝑢 to 𝑣. [5Marks] (b)Consider the statement “if Oscar eats Chinese food,then he drinks milk”. (i)Write the converse of the statement. [2Marks] (ii)Write the contrapositive of the statement. [2Marks] (c) Consider the function 𝑓:{1,2,3,4} ↦ {1,2,3,4} given by (i) Find 𝑓(1) [1Mark] (ii) Find an element 𝑛 in the domain such that 𝑓(𝑛) = 1 [1Mark] (iii) Find an element 𝑛 of the domain such that 𝑓(𝑛) = 𝑛 [1Mark] (iv) Find an element of the condomain that is not in the range [1Mark] (d)Consider the function 𝑓: ℤ ↦ ℤ given by 𝑓(𝑛) = { 𝑛 + 1: 𝑛 ⋅ 𝑒𝑣𝑒𝑛 𝑛 − 3: 𝑛 ⋅ 𝑜 ⅆⅆ (i)Is 𝑓 injective?Prove your answer [1Mark] (ii)Is 𝑓 surjective?Prove your answer [1Mark]
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