## Description

MCS-013 Solved Assignment For July 2018 Session and January 2019 Session (IGNOU BCA 2nd Semester and MCA 1st Semester)

Course Code : MCS-013

Course Title : Discrete Mathematics

Assignment Number : MCA(I)/013/Assignment/2018-19

Assignment Marks : 100

Weightage : 25%

Last Dates for Submission : 15th October, 2018 (For July Session)

15th April, 2019 (For January Session)

Note: *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. For more details, go through the guidelines regarding assignments given in the Programme Guide.*

**Question 1:
**(a) Prove by mathematical induction that Σ1?(?+1) = ?/(? + 1) (3 Marks)

(b) Make truth table for followings: (3 Marks)

i) p→ (~ q ∨ ~ r) ∧ (p ∨ r)

ii) p→(~ r ∧ q) ∧ (p ∧ ~ q)

(c) Draw a Venn diagram to represent followings: (2 Marks)

i) (A ∪ B ∩ C) ∪ (B ∩ C ∪ D)

ii) (A ∪ B ∩ C) ∩ (C~A) ∩ (A ∪ C)

(d ) Obtain the truth value of disjunction of “ Water is essential for life” and “2+2=4”. (2 Marks)

**Question 2:**

(a) Write down suitable mathematical statement that can be represented by the following symbolic properties. (2 Marks)

i) ( **∃** x) ( **∃** y) ( **∀** z) P

ii) ( **∀** x) ( **∃** y) ( **∃** z) P

(b) What are conditional connectives? Explain with example. (2 Marks)

(c) Write the following statements in the symbolic form. (2 Marks)

i) Some students can not appear in exam.

ii) Everyone can not sing.

(d) What are different methods of proof? Example with example. (4 Marks)

**Question 3:**

(a) Draw logic circuit for the following Boolean Expression: (2 Marks)

(x y z) + (x+y+z)’+(x’zy’ )

(b) What is dual of a boolean expression? Explain with the help of an example. (2 Marks)

(c) What is proper subset? Explain with the help of example. (2 Marks)

(d) What is relation? Explain properties of relations with example. (4 Marks)

**Question 4:**

(a) How many different committees can be formed of 10 professionals, each containing at least 2 Project Managers, at least 3 Team Leaders and 1 Vice President. (3 Marks)

(b) There are two mutually exclusive events A and B with P(A) =0.5 and P(B) = 0.4.

Find the probability of followings: (2 Marks)

i) A and B both occur

ii) Both A and B does not occur

(c) What is equivalence relation? Explain use of equivalence relation with the help of an example. (3 Marks)

(d) Explain the basic properties of sets. (2 Marks)

**Question 5:**

(a) How many words can be formed using letter of DEPARTMENT using each letter at most once? (2 Marks)

i) If each letter must be used,

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

(b) Show using truth table whether (P ∧ Q ∨ R) and (P ∨ R) ∧ (Q ∨ R) are equivalent or not. (2 Marks)

(c) Explain whether (P ∧ Q) → (Q → R) is a tautology or not. (3 Marks)

(d) Find dual of boolean expression for the output of the following logic circuit. (3Marks)

**Question 6:**

(a) How many ways are there to distribute 10 district objects into 4 distinct boxes with: (2 Marks)

i) At least two empty box.

ii) No empty box.

(b) Explain principle of multiplication with an example. (2 Marks)

(c ) Set A,B and C are: A = {1, 2, 3,5, 7, 9 11,13}, B = { 1,2, 3 ,4, 5,6, 7,8,9 } and C { 1,2 ,4,5,6,7,8,10, 13}. Find A **∩** B **∩** C , A **∪** B **∪** C, A **∪** B **∩** C and (B~C) (3 Marks)

(d) Show whether √11 is rational or irrational. (3 Marks)

**Question 7:**

(a) What is power set? Write power set of set A={1,2,3,4,5,6,7,9}. (2 Marks)

(b) Give geometric representation for followings: (3 Marks)

i) { -3} x R

ii) {1, -2) x ( 2, -3)

(c) Explain inclusion-exclusion principle with example. (2 Marks)

(d) Show that : (3 Marks)

(P →Q)→Q ⟹ P∨Q

**Question 8:**

(a) Explain whether function: (2 Marks)

f(x) = x2 posses an inverse function or not.

(b) What are Demorgan’s Law? Explain the use of Demorgen’s law with example. (3 Marks)

(c) Explain addition theorem in probability, with example. (2 Marks)

(d) Explain distributive laws of Boolean Algebra. (3 Marks)

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