Course Code | MCS-013 |

Course Title | Discrete Mathematics |

Assignment Number | MCA (2)/013/Assignment/2019-20 |

Maximum Marks | 100 |

Last Date of Submission | 15th October, 2019 (for July 2019 Session) 15th April, 2020 (for January 2020 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. **

**Q1. **

(a) What is proposition? Explain different logical connectives used in proposition with the help of example. **(3) **

(b) Make truth table for followings. **(4) **

- i) p→(q ∨ r) ∧p ∧~q ii) p→(~r ∨ ~ q) ∧(p ∨ r)

(c) Give geometric representation for followings: **(3) **

- i) R x { 2} ii) {1, 5) x ( -2, -3)

**Q2. **

(a) Draw a Venn diagram to represent followings: **(3) **

- i) ( A ⋂ B ∪ C) ~A ii) (A∪B∪ C) ⋂(B⋂C)

(b) Write down a suitable mathematical statement that can be represented by the following

symbolic properties. **(4) **

- i) (∃x) (∃z) ( ∀ y) P ii) ∀ (x) (∀y) (∃z) P

(c) Show whether √3 is rational or irrational. **(3) **

**Q3. **

(a) Explain inclusion-exclusion principle with example. **(2) **

(b) Make logic circuit for the following Boolean expressions: **(4) **

- i) (x’y’z) + (x’y’z)’ ii) ( x’yz) (x’yz’) (xy’z)

(c) What is a tautology? If P and Q are statements, show whether the statement

(P → Q) ∨ (Q → P) is a tautology or not. **(4) **

**Q4. **

(a) 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.
**(2)**

(b) Show that: **(2) **

P=>Q and (~P∨ Q) are equivalent.

(c) Prove that n! (n + 2) = n!+ (n +1)! **(4) **

(d) Explain principal of duality with the help of example. **(2) **

**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? **(4) **

(b) What are Demorgan’s Law? Explain the use of Demorgen’s law with example. **(4) **

(c) Explain addition theorem in probability. **(2) **

**Q6. **

(a) How many ways are there to distribute 17 district objects into 7 distinct boxes with:

- i) At least two empty box. ii) No empty box.
**(3)**

(b) Explain principle of multiplication with an example. **(3) **

(c) Set A,B and C are: A = {1, 2, 5,7,8,9,12,15,17}, B = { 1,2, 3 ,4, 5,9,10 } and

C { 2, 5,7,9,10,11, 13}. Find A ⋂B∪C , A ∪ B ∪ C, A ∪ B⋂ C and (B~C) **(4) **

**Q7. **

(a) Find how many 3 digit numbers are even? **(2) **

(b) What is counterexample? Explain how counter example helps in problem solving. **(3) **

(c) What is a function? Explain following types of functions with example.

- i) Surgective ii) Injective iii) Bijective
**(3)**

(d) Write the following statements in symbolic form:

- i) Mohan is poor but happy ii) Either work hard or be ready for poor result
**(2)**

**Q8. **

(a) Find inverse of the following function: **(2) **

f(x) = 26 2–+

*x**x *2 ≠ *x *

(b) What is relation? Explain equivalence relation with the help of an example. **(3) **

(c) Find dual of Boolean Expression for the output of the following logic circuit. **(3) **

(d) Explain distributive and complement properties of set with the help of examples. **(2) **