IGNOU MCS-13 - Discrete Mathematics

Q1. (a) What is Set? Explain use of Set with examples (b) Make truth table for followings. (c) Give geometric representation for followings: i) {5, -3) x ( -2, -2) ii) {-1, 3) x ( -2, 3) Q2. (a) Draw Venn diagram to represent followings: (b) Write down suitable mathematical statement that can be represented by the following symbolic properties. (c) Show whether √7 is rational or irrational. Q3. (a) Explain use of inclusion-exclusion principle with example. (b) Make logic circuit for the following Boolean expressions: i) (xyz) + (xyz)' + (xz'y) ii) ( x'yz) (xyz') (xy'z) (b) What is a tautology? If P and Q are statements, show whether the statement (𝑃 → 𝑄) ∨ ( →~ 𝑃) is a tautology or not. Q4. (a) How many words can be formed using letter of “EXCELLENT” using each letter at most once? i) If each letter must be used, ii) If some or all the letters may be omitted. (b) What is a relation? What are different types of relation? Explain equivalence relation with the help of example. (c) Prove that (d) What is counterexample? Explain its use with the help of an example. Q5. (a) How many different professionals committees of 8 people can be formed, each containing at least 2 Doctors, at least 2 Public Servants and 1 IT Expert from list of 7 Doctors, 6 Public Servants and 6 IT Experts? (b) A and B are mutually exclusive events such that P(A) = 1/2 and P(B) = 1/3 and P (AU B) = 1/4. What is the probability of P(A Ո B)? (c) Find how many 3 digit numbers are odd? (d) Explain whether the function f(x) = x + 1 is one-one or not. Q6. (a) How many ways are there to distribute 21 district items into 6 distinct boxes with: i) At least two empty box. ii) No empty box. (b) Explain principle of multiplication with an example. (c) Three Sets A, B and C are: A = {1, 2,3,4,5, 8,9,12,15,17}, B = { 1,2, 3 ,4,8,9, 10 } and C {1,2,7, 9, 10, 11, 13}. Q7. (a) Explain addition theorem in probability. (b) Make Pascal's triangle up to n = 6. (c) What is a function? Explain different types of functions with example. (d) Write the following statements in symbolic form: (i) Mr. X is poor but happy. (ii) Either eat healthy food or be ready for poor health. Q8. (a) Find inverse of the following functions

Q1. (a) State and explain De Morgan’s laws for Boolean algebra. (b) Make truth table for followings. (c) Give geometric representation for followings: i) {5, 5) x ( -2, -2) ii) {1, 5) x ( -2, -3) Q2. (a) Draw Venn diagram to represent followings: (b) Write down suitable mathematical statement that can be represented by the following symbolic properties. (c) Show whether √5 is rational or irrational. Q3. (a) Explain inclusion-exclusion principle with example. (b) Make logic circuit for the following Boolean expressions: i) (x'y'z') + (x'yz)' + (xz'y) ii) ( x'yz') (xyz') (x'y'z) (c) What is a tautology? If P and Q are statements, show whether the statement (𝑃 → 𝑄) ∨ ( → 𝑃) is a tautologyor not. Q4. (a) How many words can be formed using letter of PEPSUDENT using 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: Q => P and (~Q ∨ P) are equivalent. (c) Prove that (d) Explain principal of duality with the help of example Q5. (a) How many different professionals committees of 10 people can be formed, each containing at least 2 Professors, at least 3 Managers and 3 ICT Experts from list of 10 Professors, 6 Managers and 8 ICT Experts? (b) A and B are mutually exclusive events such that P(A) = 1/4 and P(B) = 2/5 and P (AU B) = 1/2. What is the probability of P(A Ո B) (c) Explain addition theorem in probability Q6. (a) How many ways are there to distribute 15 district item into 5 distinct boxes with: i) At least two empty box. ii) No empty box. (b) Explain principle of multiplication with an example (c) Three Sets A, B and C are: A = {1, 2, 8,9,12,15,17}, B = { 1,2, 3 ,4, 10 } and C {7, 9, 10, 11, 13}. Q7. (a) Find how many 3 digit numbers are even? (b) A coin is tossed n times. What is the probability of getting exactly p tails? (c) What is a function? Explain following types of functions with example. i) Surgective ii) Injective iii)Bijective (d) Write the following statements in symbolic form: (a) Shyam is rich but unhappy (b) Either do physical exercise or be ready for poor health Q8. (a) Find inverse of the following function: What is a relation? Explain equivalence relation with the help of an example. (b) Find dual of Boolean Expression for the output of the following logic circuit. (c) Explain Logical Connectives and Logical Quantifiers with the help of examples.