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# Discrete Mathematics MCS-013 Assignment SOLUTION

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 Course Code : MCS-013 Course Title : Discrete Mathematics Assignment Number : MCA(I)/013/Assignment/15-16 Maximum Marks : 100 Weightage : 25% Last Dates for Submission : 15th October, 2015 (For July 2015 Session) 15th  April, 2016 (For January 2016 Session)

There are eight questions in this assignment, which carries 80 marks. Rest 20 marks 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.

 1.  (a)   Make truth table for followings. (4 Marks) i) p→(~q ~ r) ~p ~q ii) p→(r ~ q) (~p r) (b)  Draw a venn diagram to represent followings: (3 Marks) i) (A B) (C~A) ii) (A B) (B C) (c)  Give geometric representation for followings: (3 Marks)  i)        { 2} x R

ii)      {1, 2) x ( 2, -3)

 2. (a)  Write down suitable mathematical statement that can be represented (4 Marks) by the following symbolic properties. (i)  ( x) ( y) P (ii) (x) ( y) (   z) P (b)  Show whether √15 is rational or irrational. (4 Marks) (c)  Explain inclusion-exclusion principle with example. (2 Marks) 3. (a)   Make logic circuit for the following Boolean expressions: (6 Marks) i)        (x’ y’ z) + (xy’z)’

ii)      ( x'y) (yz’) (y’z)

iii)    (xyz) +(xy’z)

(b)  What is a tautology? If P and Q are statements, show whether the               (4 Marks)
statement is a tautology or not.

 4. (a) How many different 8 professionals committees can be formed each (4 Marks) containing at least 2 Professors, at least 2 Technical Managers and 3 Database Experts from list of 10 Professors, 8 Technical Managers and  10 Database Experts? (b) What are Demorgan’s Law? Explain the use of Demorgen’s law with (4 Marks) example. (c) Explain addition theorem in probability. (2 Marks) 5. (a) How many words can be formed using letter of UNIVERSITY using (2 Marks) each letter at most once?

i)      If each letter must be used,

ii)     If some or all the letters may be omitted.

 (b)   Show that: (4 Marks) (c) Prove that n! (n + 2) = n!+ (n +1)! (4 Marks) 6.  (a) How many ways are there to distribute 20 district object into 10 (3 Marks) distinct boxes with: i)       At least three empty box.

ii)     No empty box.

 (b)  Explain principle of multiplication with an example. (3 Marks) (c) Set A,B and C are: A = {1, 2, 4, 8, 10 12,14}, B = { 1,2, 3 ,4, 5 } (4 Marks) and C { 2, 5,7,9,11, 13}. Find A B C , A   B   C, A   B   C and (B~C) 7. (a) Find how many 3 digit numbers are odd? (2 Marks) (b) What is counterexample? Explain with an example. (3 Marks) (c)  What is a function? Explain following types of functions with example (5 Marks) i) Surgective ii) Injective iii) Bijective 8. (a) Find inverse of the following function: (2 Mark) f(x) = x3 2 x   3 x 3 (b)  Explain equivalence relation with example. (2 Mark)   9

(c)    Find Boolean expression for the output of the following logic                    (3 Marks)

circuit. (d)    Prove that the inverse of one-one onto mapping is unique.                          (3 Marks)