MASTER IN COMPUTER APPLICATIONS by A0R01wt

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									Course Code                     :      CS-07
Course Title                    :      Discrete Mathematics
Assignment Number               :      MCA(4)-07/TMA/07
Maximum Marks                   :      10
Last Date of Submission         :      15th October, 2007

This is a Tutor Marked Assignment. There are two questions in this assignment.
Answer all questions. You may use illustrations and diagrams to enhance
explanations. Please go through the guidelines regarding assignments given in the
Programme Guide for the format of presentation.

Question 1:

Enumerate various steps that can be useful in translating or arguments in English to
corresponding statements or arguments in a formal language like Propositional Calculus (PC) or
First Order Predicate Calculus (FOPC). Further translate the following arguments in English to
corresponding arguments in a Formal Language (only PC/FOPC) and further check in PC/FOPC
the validity of the arguments:

(a)    (i)     If Socrates escapes and does what is wrong, then he causes his friends to suffer
               and is rightfully regarded as an outcast.
       (ii)    If Socrates causes his friends to suffer, then he does wrong.

       Therefore we conclude:

       If Socrates escapes, then he does wrong.

(b)    Lord Krishna is loved by everyone who loves someone. Further there is no person who
       does not love anybody. Therefore, conclude that Lord Krishna is loved by everyone

                                                                                       (5 Marks)


Question 2:

(a)    Let S be a non-empty set and P(S) denote the set of all subsets of S. Then show that P(S)
       alongwith the two operations, viz.,  (intersection) and  (union) forms a distributive
       lattice.

(b)    If we define a tree as a finite, undirected graph with no cycles, then prove that a graph
       with n vertices is a tree if and only it is connected and has exactly (n – 1) edges.

                                                                                       (5 Marks)
Course Code                      :        CS-07
Course Title                     :        Discrete Mathematics
Assignment Number                :        MCA(4)-07/Project/07
Maximum Marks                    :        15
Last Date of Submission          :        31st October, 2007

This is a Project Assignment. Answer all the questions. You may use illustrations
and diagrams to enhance the explanation.

Question 1:

Translate the following English arguments into propositional calculus (PC) and decide, in PC, for
each argument whether it is valid argument or not:

(a)     (i)     If belief in God has scientific backing, then it’s rational.
        (ii)    No conceivable scientific experiment could decide the issue of whether there is a
                God.
        (iii)   If belief in God has scientific backing, then some conceivable scientific
                experiment could decide the issue of whether there is a God.

                Therefore we conclude:

                Belief in God isn’t rational.

(b)     (i)     Every event with finite probability eventually takes places.
        (ii)    If the world powers don’t get rid of their nuclear weapons, then there’s a
                finite probability that humanity will eventually destroy the world.
        (iii)   If every event with finite probability eventually takes places and there’s a
                finite probability that humanity will eventually destroy the world, then humanity
                will eventually destroy the world.

                Therefore we conclude:

                Either the world powers will get rid of their nuclear weapons, or humanity will
                eventually destroy the world.

(c)     (i)     If materialism is true, then idealism is false.
        (ii)    If idealism is true, then materialism is false.
        (iii)   If mental events exist, then materialism is false.
        (iv)    If the materialist thinks that his or her theory is true, then mental events
                exist.

                Therefore we conclude:

                If the materialist thinks that his or her theory is true, then idealism is true.
                                                                                              (6 Marks)
Question 2:

(a)     The logical binary operations NAND denoted by  and NOR denoted by  are defined as
        follows:


                                                  2
                  (P  Q)  ~ (P ^ Q)
                  (P  Q)  ~ (P v Q)

                  where ~, ^ and v respectively denote the logical operations of negation,
                  conjunction and disjunction. Express
                  (i)    (P  Q) and (P  Q)  R in terms of only 
                  (ii)   (~ P  Q) in terms of only                                       (2 Marks)

(b)    Obtain Principal Disjunctive Normal and Principal Conjunctive Normal Form for the
       formula:

                  (P  Q)  (R  S)

                  where P, Q, R and S logical (Boolean) Variables.                                (2 Marks)

(c)    Determine the validity of the conclusion S  P from the following three premises:

                  ~ P v Q , ~ (Q  P v ~ R) and ~ (R  ~ S)  (R v ~ S)
                                                                                                   (1 Mark)

Question 3:

(a)    Find a minimal spanning tree for the connected weighted graph using
       (i) Kruskal’s Method (ii) Prim’s Method:


                                         B
                                                     6
                                 7               8            C
                                     5
                     A                           S       10           8
                             7
                                     D                                    E
                                                     9
                                                                                                  (2 Marks)

       (b)        Apply Dijkstra’s algorithm to find shortest path from the vertex S to vertex T.

                                             A                    4                   C

                             6                                                                5

              5                                                                           2
                                                     7                        8                    E
                         7
                                                                                              4
                                         B                    7                   D


                                                                                                  (2 Marks)



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