Discrete Structure and Optimization MCA paper Nov 2021

  • Subject Code: - PGCA 1917
  • Subject Name: - Discrete Structure and Optimization
  • Date of Examination: - Nov 2021
  • Class: - MCA 1st
  • Exam Mode: - Online

Instructions to Candidates

  1. Attempt any FIVE question(s), each question carries 14 marks.

QUESTIONS

    1. Prove the following : X ∩ (Y ⋃ Z) = (X ∩ Y) ⋃ (X ∩ Z)
    2. How many symmetric relations will there be on a set of three elements?
    2.
  2. Investigate the function f (x) = x2 + 3x + 2 for increasing and decreasing function in its entire domain.
    3.
  3. Prove that the set {0,1,2} forms a field with respect to addition and multiplication modulo 3.
    4.
    1. Simplify the following Boolean functions using the Karnaugh map: F = x'y'z' + x'yz' + xyz' + xy' z'
    2. Find the number of diagonals of a polygon having n sides.
    5.
  5. Prove that every group of prime order is cyclic.
    6.
    1. In how many ways can 5 boys and 5 girls be arranged in a round table so that 2 girls are not seated together?
    2. Solve the following recurrence relations: ar = 6ar–1 – 8ar–2, given that a0 = 4 and a1 = 10
    7.
  7. Prove that in a graph G the number of vertices having an odd degree is always even.
    8.
    1. A graph G is disconnected if and only if its vertex set V is partitioned into two nonempty, disjoint subsets V1 and V2 such that there exists no edge in G whose one end vertex is in V1 and the other is in V2.
    2. Define a regular graph and a complete graph. Draw a regular graph that is also complete.
    9.

ANSWERS:

Q1.

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