BSc IT Sem I 2019 20 2019-20 DISCRETE MATHEMATICS Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Solve any three of the following 15 marks
   • a) Check the argument is valid or invalid If | got Salary I’ll buy T.V. If I sell my Scooty I'll buy T.V. Therefore If I got salary or I sell my scooty then I ‘Il buy a
   • (b) Let U ={ 1,2,3,-------10 },X = {1,2,3,4,5 } Y ={y/y=2 x LL
   • d) In a coliege 100 students have access to 3 software packages A,B.C.28 did not use all the three packages,13 used both A and B.Draw a Venn diagram with all sets enumerated for as possible. Label the two subsets which cannot be enumerated as x
   • i)If twice as many students use package B as package A .Write down the pair of simultaneous equation in x and y.find x and y
   • ii)How many students used package C?
   • f)Simplify and show that i) (X.y)+ (x.z) +(x.p) =x
2. Q2 Solve any three of the following 15 marks
   • a) Prove that is irrational
   • b) Define Euclidian Algorithm and prove that for all real number x and fo all integer Prove that if 5n+2 is odd then n is odd dy Define Quotient Reminder theroem and prove that for all x y belong to zifx mod that the difference of any two rational number is rational number }) Prove that for all integer m and and m-nare either or odd
3. Q3 Solve any three of the following 15 marks
   • a)Show that n>3 = 1 + by strong induction
   • b)Let — be any n sets we show by Mathematical induction that
   • c) Prove that if the given predicate is true before entering loop it is true after entering loop:while 0 and m < 100)
   • d)Define Recurrisive relation and find explict formula = forall integer
   • e)Define and give suitable example of each —i)Range ii)Constant function function and Decoding function
   • f) If A ={0,1,2,3,4} and define and g:A—A such that g(x)= x? + 3x +1mod5 Does F=g?explain
4. Q4 Solve any three of the following 15 marks
   • b)Define and give counter example of each Simple graph,Pseudo graph ,circuit, Trail Degree of vertex
   • c)Prove that (z* /) is partial order relation. a
   • d)Apply prims and krushal algorithm on following digraph
   • e)If R is relation on Z defined R iff 3x+5y is divisible by that R is an
   • f)Check the following graphs are isomorphic or not.Define Hand ShakingTheroem
5. Q5 Solving any three of the following. 15 marks
   • a)Calculate the expected value of X sum of the scores when two dices are rolled
   • b)A factory has two machines.The empirical evidance has established that machines produce 30% and 70% output has also been established that 5% and 1% produced by these machines respectively was defective.A defective item drawn at random. What is the probability that defective item was produced by machine
   • c)Use Pascals formula to prove by mathematical induction that ifn is an integer and
   • d)Consider two events A and B such that p(A) =1/8 ,p(A/B) =1/4 p(B/A) =1/6 Examine the folowing statement and comments on validity of each of these i)A and B ARE independent ii)A and B are mutually exclusive iii) Occarance of A implies that
   • e)State Pigeon —Hole Principal Show that if seven numbers from Ito 12 are selected then at least two of them will add thirteen four vowels and eight consonants i) find the number of five letterswords containing two different vowels and three different consonants.ii)Find the number of five letter words if the wordsmust contain the letter B.iii)Find the number of five letter words ii the words must begin with B iv) Find the number of five ictter words if they begin with A and contain B

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