**UNIVERSITY EXAMINATIONS 2017/2018**

**ORDINARY EXAMINATION FOR BACHELOR OF SCIENCE INFORMATION**

**TECHNOLOGY, BSc. APPLIED COMPUTING **

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

**DATE: DECEMBER 2018 TIME: 2 HOURS**

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

**QUESTION ONE**

a) Find the sum of products for the function πΉ(π₯, π¦, π§) = (π₯ + π§)π¦Μ
using

Boolean identities (4 Marks)

b) Find the positive integer π such that 10 C n =

10 C n + 2 (6 Marks)

c) i) Distinguish between a multigraph and a pseudograph. (2 Marks)

ii) How many edges are there in a graph with 10 vertices each of degree six? (2 Marks)

d) Consider the directed graph G below

i) Find the indegree and outdegree of each vertex (2 Marks)

ii) Find the successor list of each vertex G (2 Marks)

iii) Are there any sources or sinks? (2 Marks)

e) Solve the following linear homogeneous recurrence relation:

ππ = 4 ππβ1 – 4 ππβ2 with initial conditions

π0 = 4

π1 = 12 (2 Marks)

2

f) Given the Sets:

A = {3, 5, 7, 8, 11, 12}

B = {2, 5, 6, 8, 10, 12}

Determine A β© B (2 Marks)

g) Show that p Λ q logically implies p β q (3 Marks)

h) Consider a graph where

V (G) = {A, B, C,D} and

E (G) = [{A, B,}, {B, C}, {B,D}, {C,D}]

Find the degree and parity of each vertex in G (3 Marks)

TOTAL: [30 MARKS]

**QUESTION TWO**

a) Draw the multigraph G whose adjacency matrix A is A =

b) Let G be the graph where

V (G) = {A, B, C, X, Y, Z} and

E (G) = [{A, C}, {A, X}, {A, Y}, {B, Y}, {B, Z}]

i) Find G-A (3 Marks)

ii) Determine the number of connected paths of G-A (3 Marks)

c) Draw the Diagrams of each of the following graphs G (V,E):

i) V = {A,B,C,D}

ii) E = [{A,B},{A,D},{B,C},{B,D}{C,D}] (3 Marks)

iii) V= {a, b, c, d, e}

iv) E= [{a, b}, {a, c}, {b, c}, {d, e}] (3 Marks)

d) Order the sets of numbers: Z, IR,QI , IN using β (3 Marks)

e) Consider the propositions:

p: Juan is a math major.

f: Juan is a computer science major.

Use symbolic connectives to represent the proposition βJuan is a math major but not a

computer science major.β (3 Marks)

TOTAL [20 MARKS]

**QUESTION THREE**

a) Define the following terms

i) One-to-one (or injective ) Functions (2 Marks)

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

iii) One-to-one correspondence (or bijective) Functions (2 Marks)

iv) Invertible Functions (2 Marks)

b) Let A=

a,b,c,d,e

and B be the set of letters in the alphabet. Let the functions f, g and h

from A into B be defined as follows:

f g h

a β z a β z a β z

b β a b β y b β c

c β s c β x c β e

d β r d β y d β r

e β e e β z e β s

Are any of these functions one to one? (2 Marks)

c) Consider the functions

f(x) = 2x β 3 and g(x) = x

2 + 3x + 5

Find a formula for the composition functions:

i) g β f (1 Mark)

ii) f β g (2 Mark)

d) Consider the maps

f βΆ A β B, g βΆ B β A

h βΆ C β B, F: B β C

G βΆ A β C

i) Is gβf defined? If so what is the domain and codomain? (2 Marks)

ii) Is hβf defined? If so what is the domain and codomain? (2 Marks)

iii) Is FβhβG defined? If so what is the domain and codomain? (2 Marks)

Is GβFβh defined? If so what is the domain and codomain? (2 Marks)

TOTAL: [20 MARKS]

**QUESTION FOUR**

a) List the elements of the following sets

i) A = {x: x β N, 3 < π₯ < 9} (1 Mark)

ii) B = {x: x β N, x

2 + 1 = 10} (1 Mark)

iii) C = {x: x β N, x is odd , β5 < π₯ < 5} ` (1 Mark)

b) (i) Draw a Venn diagram of sets A, B, C where A β B, sets A and C are disjoint

but B and C have elements in common (4 Marks)

(ii) Write A β© (B βͺ C) as the (disjoint) union of fundamental products. (5 Marks)

c) 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) How many committees of five with a given chairperson can be selected from twelve persons?

(5 Marks)

(b) How many diagonals has a regular polygon with n sides (5 Marks)

(c) Which regular polygon has the same number of diagonals as sides (5 Marks)

(d) Prove P(n,r) =

π!

(πβπ)!

= π(π β 1)(π β 2) β¦ . . (π β π + 1) (5 Marks)

TOTAL: [20 MARKS]