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    Jntu Kakinada - M.tech - Mathematical Foundations of Computer Science Sup FR 28

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    prahari guduru
    This document contains 8 questions regarding mathematical foundations of computer science for a Master's of Technology supplementary examination. The questions cover topics such as logic, sets, functions, relations, counting, recurrence relations, graphs, modular arithmetic, and binomial coefficients. Students are asked to solve problems, prove statements, find relations, and differentiate concepts.

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    Jntu Kakinada - M.tech - Mathematical Foundations of Computer Science Sup FR 28

    Uploaded by

    prahari guduru
    36 views2 pages
    This document contains 8 questions regarding mathematical foundations of computer science for a Master's of Technology supplementary examination. The questions cover topics such as logic, sets, functions, relations, counting, recurrence relations, graphs, modular arithmetic, and binomial coefficients. Students are asked to solve problems, prove statements, find relations, and differentiate concepts.

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    Jntu Kakinada_m.tech_mathematical Foundations of Computer Science Sup Fr 28_firstranker.com

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    This document contains 8 questions regarding mathematical foundations of computer science for a Master's of Technology supplementary examination. The questions cover topics such as logic, sets, functions, relations, counting, recurrence relations, graphs, modular arithmetic, and binomial coefficients. Students are asked to solve problems, prove statements, find relations, and differentiate concepts.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    36 views2 pages

    Jntu Kakinada - M.tech - Mathematical Foundations of Computer Science Sup FR 28

    Uploaded by

    prahari guduru
    This document contains 8 questions regarding mathematical foundations of computer science for a Master's of Technology supplementary examination. The questions cover topics such as logic, sets, functions, relations, counting, recurrence relations, graphs, modular arithmetic, and binomial coefficients. Students are asked to solve problems, prove statements, find relations, and differentiate concepts.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

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    com

    Code No: G0501/R13


    M. Tech. I Semester Supplementary Examinations, December-2016
    MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
    (Common to CS and CS&E)
    Time: 3 hours Max. Marks: 60
    Answer any FIVE Questions
    All Questions Carry Equal Marks

    1. a Use De Morgan’s laws to find the negation of each of the following statements. 6M
    (a) Jan is rich and happy.
    (b) Carlos will bicycle or run tomorrow.
    (c) Mei walks or takes the bus to class.
    (d) Ibrahim is smart and hard working.
    b Using truth tables, 6M
    (a) Show that p ↔ q and (p ˄q) ˅(¬p ˄ ¬q) are logically equivalent.
    (b) Show that (p ˄ q) → r and (p → r) ˄(q → r) are not logically equivalent.

    2. a Let f be a function from the set A to the set B. Let S and T be subsets of B. 6M
    We denote the inverse image of S by f-1 (S), so that f-1 (S) = {a
    om

    ∈A/f(a) ∈ S}. Show that (i) f-1 (S ∪ T) = f-1 (S)∪ f-1 (T)
    (ii) f-1 (S ∩ T) = f-1 (S)∩ f-1 (T)
    r.c

    b For each of the following relations on the set {1, 2, 3, 4}, list the properties of each 6M
    ke

    relation. (reflexive, symmetric, antisymmetric, irreflexive, asymmetric, or transitive)


    an

    a. {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}
    b. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}
    tR

    c. {(2, 4), (4, 2)}


    irs

    d. {(1, 2), (2, 3), (3, 4)}


    .F

    e. {(1, 2), (2, 2), (3, 3), (4, 4)}


    w

    f. {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
    w
    w

    3. a i. How many bit strings are there of length six or less? 6M


    ii. How many strings of four decimal digits
    (a) do not contain the same digit twice?
    (b) end with an even digit?
    (c) have exactly three digits that are 9s?
    iii. How many numbers must be selected from the set {1; 3; 5; 7; 9; 11; 13; 15}
    to guarantee that at least one pair of these numbers add up to 16?
    b Discuss the principles of Inclusion – Exclusion & give its applications. 6M

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    Code No: G0501/R13

    4. a Solve an – 3 an-1 = 2, n ≥2, with a0 =1. 6M


    b Find the reccurence relation with initial condition for the following: 6M
    (i) 2, 10, 50, 250, ……. (ii) 1, 1, 3, 5, 8. 13. 21, …..

    5. a i. Define Eulerian Graph and prove that a non empty connected graph G is 6M
    Eulerian if and only if its vertices are all of even degree.
    ii. Differentiate between Eulerian graph & Hamiltonian graph with example.
    b Find the Minimal Spanning Tree for the following graph using Kruskals & Prims 6M
    Algorithm .

    6. a Let Q(x,y) denote "x + y = 0". What are the truth values of the quantifications 6M
    om

    ∃y∀xQ(x,y) and ∀x∃yQ(x,y), where the domain for all variables consists of all real
    numbers?
    r.c

    b Show that R is logically derived from P → Q, Q → R, and P 6M


    ke

    7. a Prove that the relation “congruence modulo m” over the set of positive integers is an 6M
    an

    equivalence relation
    tR

    b Draw the Hasse Diagram for the partial ordering set (D (128), ≤) and ≤ is the 6M
    irs

    relation “is a Divisor of”.


    .F
    w

    8. a Solve the function using recursive substitution f(n)=f(n/2)+1;f(1)=1. 6M


    w

    b Show that : i)nC0+nC1+nC2+nC3+…..ncn=2n 6M


    w

    ii)nC0-nC1+nC2-+……(-1)nnCn=0.
    *****

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