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Course Code : MCS-013
Assignment Number : MCA(1)/013/Assign/06
Maximum Marks: 100
Weightage : 25%
Last Date of Submission : 15^{th} April, 2006
There are eight questions in this assignment. Answer all questions. 20 Marks are for viva-voce. 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.
Q1: (a) Give the truth tables for the following:
i) q -> (p ~ r) ^p ^q
ii) p -> q ~ q ^p ^r (4 Marks)
(b) Explain with example, the use of conditional connectives in forming the prepositions. (2 Marks)
(c) Write the suitable mathematical statements that can be represented by the following symbolic properties:
i) ( x) ( y) P
ii) (x) ( y) ( z) P (4 Marks)
Q2: (a) Explain different methods of proof with the help of an example for each. (6 Marks)
(b) Show whether root 11 is rational or irrational. (4 Marks)
Q3: (a) Define Boolean algebra. Also explain with an example, how Boolean algebra methods are used in circuit design. (5 Marks)
(b) If p and q are statements, show whether the statement [(p? ~q) (~ q)] ? (p) is a tautology or not. (5 Marks)
Q4: (a) Make logic circuit for the following Boolean expressions:
i) (x.y + z) + (x+z)'
ii) x'.y'+ y.z'+z'.x' + x .y(4 Marks)
(b) Find the boolean expression for the output of the following logic circuit: (4 Marks)
(c) Write a superset for the sets.(2 Marks)
A = {1, 2, 3, 4, 9}, B = {1,2 } and C { 2, 5,11}
Q5: (a) Draw a Venn diagram to represent following: (4 Marks)
i) (A B) n (C~A)
ii) (A u B) n (B n C)
b) Give geometric representation for following (4 Marks)
i) { 3, 2} x R
ii) {1, -2) x ( -2, -3)
(c) What is principle of strong mathematical induction? In which situation this principle is used. (2 Marks)
Q6: (a) What is a permutation? Explain circular permutation with example. (6 Marks)
(b) Find inverse of the following functions: (4 Marks)
i) f(x) = x^{2} + 5 / x - 3, x not equal to 3
ii) f(x) = x^{3} - 8 / x - 2, x not equal to 2
Q7: (a) How many 4 digits number can be formed from 6 digits 1, 2, 3, 4, 5, 6 if repetitions are not allowed? How many of these numbers are less than 3000? How many are odd? (4 Marks)
(b) How many different 7 persons committees can be formed each containing at least two women and at least one man from a set of 10 women and 15 men. (4 Marks)
(c) Explain partition (integer partition) with an example. (2 Marks)
Q8: (a) What is a pigeon hole principle? Explain the applications of pigeonhole principle with example? (4 Marks)
(b) How many ways are there to distribute or district object into 12 distinct boxes with
i) At least two empty box.
ii) No Empty box. (4 Marks)
(c) In how many ways 20 experts of same technologies can be placed in 7 same kinds of expert committees. (2 Marks)
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