University : Teerthanker Mahaveer University
College : Teerthanker Mahaveer College of Management & Computer Applications
Degree : MCA
Subject : MCA103 Discrete Mathematics
Semester : I
Document Type : Question Paper
Website : tmu.ac.in
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Discrete Mathematics :
MCA I SEMESTER EXAMINATION 2010-11
Course Code : MCA103
Paper ID: 0491102
Related : Teerthanker Mahaveer Medical College & Research Centre MCA102 Environmental Science & Ethics MCA Question Paper : www.pdfquestion.in/5793.html
Time : 3 Hours Max. Marks: 75
Note : Attempt six questions in all. Q. No. 1 is compulsory.
1. Answer any five of the following (limit your answer in 50 words). (3×5=15)
a) Define logical equivalence. Show that ~(pvq) and (~p) (~q) are logically equivalent.
b) Define tautology. Show that the proposition p ( p q) is a tautology.
c) Define Finite, Infinite and Null sets along with one example of each.
d) Define composition of relations.
e) Define one to one function along with example.
f) Show that the function f(x) = x3 and g(x) = x1/3 for all xR are inverses of one another.
g) Find the generating function for the sequence 1, a, a2, ………, where a is a fixed constant.
h) Define finite and infinite graphs.
2.a) Explain conditional and Bi-conditional statements along with their truth tables. (6)
b) Construct a truth table for the compound proposition ( p q) (~ q r) . (6)
3.a) Write a short note on Fallacies of arguments. (6)
b) Prove that for sets A,B,C : ( A(B C) = (A B)(AC) (6)
4.a) Define carterian product of sets. If A = {1,4}, B = {4,5} , C = {5,7}, verify that A× (B C) = (A× B) (A×C) . (6)
b) If A, B, C be any three sets, then using Venn diagram, prove that A – (B C) = (A – B) (A – C) (6)
5.a) Write short note on : (3+3)
i) Equivalence relation ii) Hass diagram
b) If f : A B and g : B C be one to one onto function; then prove that (gof)-1 = f-1 og-1 (6)
6.a) Explain Injective, Subjective and Bijective functions. (6)
b) There are four roads from city X to Y and five roads from city Y to Z, findi) how many ways is it possible to travel from city X to Z via Y.
ii) How different round trips routes are there from city X to Y to Z to Y and back to X.
7.a) Solve the recurrence relation fn = fn-1 + fn-2, n 2 with the initial condition f0=1, f1 = 1. (6)
b) Use generating functions to solve the recurrence relation : (6) an+2 – 2an+1 + an = 2n;a0 = 2, a1 = 1
8. Explain the following : (3×4=12)
a) Directed and undirected graphsb) Connectivity
c) Isolated and pendent vertexd) Spanning Tree
MCA I (First) Semester Examination 2014-15 :
Course Code :MCA103
Paper ID : 0871203
Time : 3 Hours
Note : Attempt six questions in all. Q. No. 1 is compulsory.
1. Answer any five of the following (limit your answer to 50 words). (4×5=20)
a) Define Proposition and Truth Table.
b) Define Pigeon-Hole Principle.
c) Draw the complete graph K5 and K6.
d) State Hand Shaking lemma.
e) Show that K5 is non-Planar.
f) State Kuratowski’s Theorem.
g) Define Recurrence Relation.
h) Obtain the generating function of the numeric function ar = 3r+2, r ≥ 0.
2. Show the following by truth table
3. Find the number of primes not exceeding 100. (10)
4. The sum of the degree of all vertices in a graph G is equal to twice the number of edges in G. (10)
5. A connected graph G is an Eulerian graph iff all vertices of g are of even degree. (10)
6. Let G be a connected planar simple graph with v vertices and e edges and let r be the number of regions in a planar representation of G. Then r = e – v + 2. (10)
7. Define Kruskal’s and Prime’s Algorithm. (10)
8. Solve the difference equation.
MCA I (First) Semester Examination 2015-16 :
1. Answer any five of the following (limit your answer to 50 words). (4×5=20)
a) Define a graph.
b) Define planar and draw K4 planar graph.
c) What do you understand by “Contradiction” in Propositional Calculus?
d) Prove that the number of permutations of n things taken all at a time is n!.
e) Define inorder, preorder and postorder tree traversal techniques.
f) What do you understand by valid arguments and fallacy arguments?
g) Define linear recurrence relations. Also differentiate between homogeneous and non-homogeneous recurrence relations.