**UNIVERSITY EXAMINATIONS 2017/2018**

**ORDINARY EXAMINATION FOR BACHELOR OF SCIENCE**

**INFORMATION TECHNOLOGY – APPLIED COMPUTING **

**BIT 1206 DISCRETE MATHEMATICS (DAY/EVENING)**

**DATE: DECEMBER, 2017 TIME: 2 HOURS**

**INSTRUCTIONS: Answer question ONE and Any other TWO questions**

**QUESTION ONE**

(a) Let + and ∗ be operations in B = 0,1 defined in Fig 1 below. Suppose that the

compliments are defined by I = 0 and 0 ꞌ

ꞌ

= 1 show that B = 0,1 is a Boolean algebra

Find

(i) A∪B

(3 Marks) (ii) B∪C

(3 Marks)

Total [30 Marks]

**QUESTION TWO**

a) Define the following terms

(i) one-to-one (or injective ) Function (2 Marks)

(ii) Onto (or surjective) Function (1 Mark)

(iii) one-to-one correspondence (or bijective) Function (2 Marks)

(iv) Invertible Function (2 Marks)

(b) Let A= a,b,c,d,e and B be the set of letters in the alphabet. Let the functions f and g and

h

from A into B be defined as follows

(i) Is gof defined ? if so what is the domain and codomain ? (2 Marks)

(ii) Is hof defined? if so what is the domain and codomain ? (2 Marks)

(iii) Is FohoG defined ? if so what is the domain and codomain ? (2 Marks)

(iv) Is GoFoh defined ? if so what is the domain and codomain ? (2 Marks)

Total[20 Marks]

**QUESTION THREE**

(a) Define the following terms as used in graph theory

(i) A path (2 Marks)

(ii) A simple path (2 Marks)

(iii) A closed path (2 Marks)

(b) Draw a diagram for each of the following graphs

**QUESTION FOUR**

(a) List the elements of the following sets

(d) A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see

which of the three popular options, air-conditioning (A), radio (R), and power windows (W) were

already installed. The survey found

15 had air-conditioning

12 had Radio

5 had air-conditioning and Power windows

9 had air-conditioning and radio

4 had radio and power window

3 had all the three options

2 had no options

Find the number of cars that had

(i)Only power windows (2 Marks)

(ii) Only air-conditioning (1 Mark)

(iii) Only radio (1 Mark)

(iv) radio and power windows but not air – conditioning (1 Mark)

(v) air conditioning and radio but not power windows (1 Mark)

(vi) Only one of the options (2 Marks)

Total [20 Marks]

**QUESTION FIVE**

(a) Test the validity of the following argument:

S1 : If two sides of a triangle are equal, then the opposite angles are equal.

S2 : Two sides of a triangle are not equal

S: The opposite angles are not equal. (4 Marks)

(b) Determine the truth values of each statement

(i) 2 + 2 = 4 iff 3+6 = 9

(ii) 2+ 2 = 7 if and only if 5 + 1 = 2

(iii) 1 + 1 = 2 iff 3 + 2 = 8

(iv) 1 + 2 = 5 if and only if 3+ 1 = 4 (4 Marks)

(c) Use the Laws of algebra of propositions to prove the following