Discrete Mathematics Answers

Determine truth value for this statement if the domain consist of all integers

Vn( n+1>n)

A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistryand 23 took mathematics, 9 students took both chemistry and mathematics, 6 student took both physics and mathematics and all those who took physics also took chemistry. One student fell ill and failed to write any of the papers.

I. Represent the information on a Venn diagram

II. How many students took all three subjects?

III. How many students took exactly one of the subject?

IV. How many students took exactly two of the subjects?

V. How many students prepared for the examination?

Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 8},

B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations(computer representation) for A, B, and C together with UNION, intersection, difference, and complement to find the bit representation for the following:

(a) AU B

(b) An B n C

(f)) (AU C) n B

(d) (A - B) UC

(e) An (B - (C n B))

(f) A - (B - C)

(g) (AU B) U (C - B)

Develop a digital circuit diagram that produces the output for the following logical expression when the input bits are A, B and C i. (A ∧ B) ∨ ((B ∧ C) ∧ (B ∨ C)) [4 marks] ii. (A ∧ B ∧ C) ∨ A ∧ (￢ B ∨ ￢C) [4 mark

 Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the letters p, q, r and logical connectives. (i) If the flag is set, then I = 0 [2 marks] (ii) The flag is set and I = 0 if subroutine S is not completed [2 marks] (iii) Subroutine S is completed if and only if I = 0 and flag is set [2 marks] b) State the converse, contrapositive, and inverse of each of these conditional statements. (i) If it snows tonight, then I will stay at home [3 marks] (ii) I go to the beach whenever it is a sunny summer day [3 marks] (iii) When I stay up late, it is necessary that I sleep until noon [3 marks] c) Explain the step-by-step procedure involved in finding the inverse of an n by n square matrix. 

1. Let W ={1,2,….,8} Q = {2,4,6,8,10}, Y = {1,2,4,5,6,8,9}. Evaluate:
2. W union Y (2 marks)
   
3. Q intersection Y (2 marks)
   
4. Set Difference P minus Y (2 marks)
   
5. (W intersection Q) union Y

1. Out of 300 students taking discrete mathematics, 60 take coffee, 27 take cocoa, 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and cocoa, 3 take chocolate, tea and cocoa, 20 take cocoa only, 2 take tea, coffee and chocolate, 30 take coffee only, 9 take tea and chocolate whereas 12 take tea and coffee.
2. Express this information on a Venn diagram. (7 marks)
3. Find how many take any beverage. (2 marks)
4. Find how many take Fanta. Why? 

Solve the Recurrence relation a_n-3a_n-1-4a_n-2=4*3ⁿ where a₀=1,a₁=2

Solve the Recurrence relation a_n-3a_n-1-4a_n-2=4.3n where a₀=1,a₁=2