Course Title : Discrete Mathematics
Assignment Number : MCA(1)/013/Assign/2014-15
Assignment Marks : 100
Weightage : 25%
Last Dates for Submission : 15th October, 2014 (For July 2014 Session)
15th April, 2015 (For January 2015 Session)
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There are eight questions in this assignment, which carries 80 marks. Rest 20marks are for viva-voce. Answer all the questions. You may use illustrations and diagrams to enhance the explanations. Please go through the guidelines regarding assignments given in the Programme Guide for the format of presentation.
Question 1
a) Make truth table for
i) p¨(q ~ r) ( ~p ~ q )
ii) ~p¨(~r q ) (~p r)
(4 Marks)
b)
If A = {1, 2, 3, 4, 5,6,7,8, 9} B = {1, 3, 5, 6, 7, 10,12,15}and
C = {1, 2,3, 10,12,15, 45,57} Then find (A B) C.
(2 Marks)
c)
Write down suitable mathematical statement that can be represented
by the following symbolic properties.
i) ( x) ( y) ( z) P
ii) ( x) ( y) ( z) P
(4 Marks)
Question 2
a)What is proof by mathematical induction? Show that for integers greater than zero: 2n >= n+1.
b) Show whether 17 is rational or irrational. (3 Marks)
c)
Explain concept of function with the help of an example. What is relation ? Explain following types of relation with example:
i) Reflexive
ii) Symmetric
iii) Transitive
Question 3
a)A survey among the players of cricket club, 20 players are pure batsman,10 players are pure bowler, 40 players are all rounder, and 3 players are wicket keeper batsman. Find the following:
i) How many players can either bat or bowl?
ii) How many players can bowl?
iii) How many players can bat?
b) If p and q are statements, show whether the statement p¡÷q) q)] ¡÷ (~p ~q) is a tautology or not.
Question 4
a)Make logic circuit for the following Boolean expressions:
i) (x¡¬ y z) + (x y z)¡¬
ii) ( x' y) (y¡¬ z) (y z¡¬)
iii) (x y) (y z)
b)Explain principle of duality. Find dual of Boolean expression of the output of the following Boolean expression: ( x' y z) (x y¡¬ z) ¡¬ (x y z¡¬)
Question 5
a)Draw a Venn diagram to represent following:
i) (A B) (C~B)
ii) (A B) (B C)
b) if f(x) = log x and g(x) = ex, show that (fog)(x) = (gof)(x).
c) Explain inclusion-exclusion principle with example.
Question 6
a) What is pigeonhole principle? Explain its application with the help of an example.
b)If f : R „³ R is a function such that f (x) = 3x2 + 5, find whether f is one - one onto or not. Also find the inverse of f.
Question 7
a) Find how many 4 digit numbers are odd? (2 Marks)
b)How many different 10 professionals committees can be formed each containing at least 2 Project Delivery Managers, at least 2 Technical Architects and 3 Security Experts from list of 10 Project
Delivery Managers 12 Technical Architects and 5 Security Experts?
c)Explain concept of permutation with an example. How it is different from combination, explain with an example?
Question 8
a) What is Demorgan.s Law for Boolean algebra? Explain its application with example.
b)How many .words. can be formed using letter of STUDENT using each letter at most once:
i) If each letter must be used,
ii) If some or all the letters may be omitted.
c) Show whether ( p¡÷q) ( q ¡÷ p ) is a tautology or not using truth table.
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