Discrete Mathematics Answers

a) A three digit number is to be formed using the digits 1, 2, 3, 4, 5, 6 and no repetition is

allowed.

i) How many numbers can be formed if the leading digit is 4?

ii) How many numbers can be formed if the number is more than 250?

iii) How many odd numbers can be formed between 200 and 400?

(8 marks)

b) Consider a bookshelf contains 28 books in different genre. 14 books are in education, 9

books in business and 5 books in motivation. A student would like to take 15 books. Find the number of ways if:

I) there is no restriction

ii) the choice must consist of 8 books in education, 5 books in business and 2 books in

motivation genre.

iii) The choice must consist of at least 9 books in education and exactly 5 books in

motivation genre.

(7 marks)

Draw a graph having the given properties:

1. Simple graph : 6 vertices with degrees 2,3,3,3,4,5

2. Simple graph: 9 vertices with degrees 1,1,3,3,4,4,5,5,6

3. Simple graph: 7 vertices with degrees 3,3,3,3,3,3,6

4. Simple graph: 5 vertices with degrees 4,4,4,4,4

5. Simple graph: 6 vertices with degrees 1,2,3,4,5,5

6. Simple graph: 5 vertices having degrees 2,2,4,4,4

7. Simple graph : 8 vertices with degrees 2,2,3,3,3,4,4,7

8. Simple graph: 8 vertices with degrees 2,3,3,4,4,4,5,5

Let R₁ and R₂ be the relations on { 1, 2, 3, 4 } given by

  R₁ = { (1,1), (1,2), (3,4), (4,2) }

  R₂ = { (1,1), (2,1), (3,1), (4,4), (2,2) }

  List the elements of R₂ Ο R₁

Answer as required.

1. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find  (a) A U B  (b) A – B

2. Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find   (a) A ∩ B     (b) B – A

3. Let C = {1, a , 2, c}. Find  (a) Cardinality of C   (b)  P(C)

4. Consider sets in #s 2 and 3, show that     

 (a) A U (B ∩ C) = (A U B) ∩ (A U C)  (b) (A U B)c = Ac ∩ Bc.

5. Suppose that the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Express each of these sets with bit strings where the ith bit in the string is 1 if i is in the set and 0 otherwise.  

 (a) {3,4,5}

 (b) {2, 3, 4, 7, 8, 9}