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# MCS-013 Solved Assignment2021-22

### MCS-013 Solved Assignment 2021-22

 Subject Name Discrete Mathematics Assignment Code MCS-013 Session 2021-2022 (July – January) File Type PDF Number of Pages 42 Price Rs. 45

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#### Discrete Mathematics (MCS-013) 2021-2022 (July-January)

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. Please go through the guidelines regarding assignments given in the Programme Guide for the format of presentation.

Question1:
(a) What is proposition? Explain different logical connectives used in proposition with the help of example. (3 Marks)

(b) Make truth table for followings. (4 Marks)
i) p→(q ∨ r) ∧ p ∧ ~q
ii) p→(~r ∨ ~q) ∧ (p ∨ r)

(c) Give geometric representation for followings: (3 Marks)
i) R × { 3}
ii) {1, 5) × ( 3, − 2)

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Question2:
(a) Draw a Venn diagram to represent followings: (3 Marks)
i) (A ∪ B $\small&space;\cap$ C) ∪ (A ∪ B $\small&space;\cap$ C)
ii) (A ∪ B $\small&space;\cap$ C) $\small&space;\cap$ (A $\small&space;\cap$ B ∪ C)

(b) Write down suitable mathematical statement that can be represented by the following symbolic properties. (2 Marks)
i) (∃x) (∀y) (∀z) Q
ii) (∀z) (∀y) (∃z) P

(c) Show whether √5 is rational or irrational. (3 Marks)

(d) Explain circular permutation with the help of an example. (2 Marks)

Question 3:
(a) Make logic circuit for the following Boolean expressions: (6 Marks)
i) (x’ y z’) + (xyz)’

ii) (x’y) (y’z’) (yz’)

iii) (xyz) + (x’y’z)

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

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Question 4:
(a) How many different committees of 6 professionals can be formed, if each committee contains at least 1 Professor, at least 2 Technical Managers and 1 Database Experts from list of 6 Professors, 5 Technical Managers and 8 Database Experts? (4 Marks)

(b) What are Demorgan’s Law? Explain the use of Demorgen’s law with the help of example. (4 Marks)
(c) Explain addition theorem in probability. (2 Marks)

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Question 5:
(a) How many ways are there to distribute 15 district objects into 5 distinct boxes with: (3 Marks)
i) At least three empty box.
ii) No empty box.

(b) Find how many 3 digit numbers are even? (3 Marks)

(c) Set A,B and C are: A = {1, 2, 3, 4, 5, 7, 8, 9, 11, 17}, B = {1, 2, 3 , 4, 5, 9, 11, 12} and C ={2, 3, 5, 7, 9, 10, 11, 12, 13}.

Find A $\small&space;\cap$ B ∪ C , A ∪ B ∪ C, A ∪ B $\small&space;\cap$ C and (B~C) (4 Marks)

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Question 6:
(a) How many words can be formed using letter of TEACHER using each letter at most once? (3 Marks)
i) If each letter must be used,
ii) If some or all the letters may be omitted.

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

(c) Prove that 12 + 22 + 32 + …+ n2= n(n+1)(2n+1)/6 ; ∀n ∈ N (4 Marks)

Question 7:
(a) What is principle of duality? Explain with example. (3 Marks)

(b) What is power set? Write power set of set A={1, 2, 3, 4, 7, 9, 11}. (3 Marks)

(c) What is a function? What is domain and range of a function? Explain with example. (4 Marks)

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Question 8:
(a) Find inverse of the following function: (3 Marks)

$\small&space;f\left&space;(&space;x&space;\right&space;)=\frac{x^{3}+2}{x-3}\;&space;\;&space;x\neq&space;3$

(b) Explain equivalence relation with example. (3 Marks)

(c) Prove that the inverse of one-one onto mapping is unique. (2 Marks)

(d) What is counterexample? Explain with an example. (2 Marks)

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