**UNIVERSITY EXAMINATIONS: 2012/2013**

**FIRST YEAR EXAMINATION FOR THE BACHELOR OF SCIENCE **

**IN INFORMATION TECHNOLOGY**

**BIT 1206 DISCRETE MATHEMATICS **

**DATE: AUGUST, 2013 TIME: 2 HOURS**

**INSTRUCTIONS: Answer Question ONE and any other TWO Questions**

**QUESTION ONE**

a) Consider the following data for 120 mathematics students at a college concerning

the languages French, German and Russian:

65 study French, 45 study German, 42 study Russian, 20 study French and

German, 25 study French and Russian, 15 study Russian and German and 8 study

all three languages.

Find the number of students who study at least one of the three languages.

(5 Marks)

b) A man, woman, boy, girl, dog and cat are walking down a long and winding road

one after the other.

i) In how many ways can this happen? (2 Marks)

ii) In how many ways can this happen if the dog comes first. (3 Marks)

(5 Marks)

d) Determine the validity of the following argument:

If my brother passes the CPA exams in the first attempt, l will give him a valuable

prize. Either he passes the exams or l was out of the country. I did not give my

brother a prize this time although he passed the exam. Therefore, l was out of the

country. (5 Marks)

**QUESTION TWO**

a) Let

p q

be “If it rains, then l get wet”. Write down the inverse, converse and

contrapositive of this proposition. (6 Marks)

b) Write the negation of the following statements:

i) He swims if and only if the water is warm.

ii) Kenya will be prosperous if and if its citizens are industrious.

iii) If it is cold , then he wears a coat but no sweater. (6 Marks)

c) Construct the truth table for

**QUESTION THREE**

a) Find the smallest positive integer x such that when x is divided by 3 it yields a

remainder 2, when x is divided by 7 it yields a remainder 4, and when x is divided

by 10 it yields a remainder 6. (8 Marks)

**QUESTION FOUR**

a) Define a Boolean Algebra. (5 Marks)

b) Write the dual of each Boolean equation:

(5 Marks)

d) Test the validity of the following argument: If the rainfall is good, the yield of

crops is good. The yield of crops was not good. Therefore the rainfall was not

good. (6 Marks)

**QUESTION FIVE**

a) Draw the graph K2,5. (5 Marks)

b) Which connected graphs can be both regular and bipartite? (5 Marks)

c) Of a group of 20 persons, 10 are interested in music, 7 are interested in

photography and 4 like swimming; further more 4 are interested in both music

and photography, 3 are interested in both music and swimming, 2 are interested in

both photography and swimming and one is interested in music, photography and

swimming. How many are interested in photography but not in music and

swimming? (5 Marks)