**UNIVERSITY EXAMINATIONS: 2018/2019**

**EXAMINATION FOR THE DIPLOMA IN BUSINESS INFORMATION**

**TECHNOLOGY**

**DBIT103 FOUNDATION OF MATHEMATICS**

**DATE: NOV – DEC 2018 TIME: 2 HOURS**

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

**QUESTION ONE**

a) Construct a truth table for (~ p ∨ q) ∧ ( p → ~ q) (5 Marks)

b) Define the following terms as used in propositional logic

i) Contradiction

ii) Contingency

iii) Tautology (9 Marks)

c) Given the Sets:

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

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

C= {2, 4, 5, 6, 9, 11, 12},

Determine the following sets

i) A ∩ B ∩ C (2Marks)

ii) A ∩ B (2Marks)

iii) A ∩ C (2Marks)

iv) B ∩ C (2Marks)

Hence represent the sets obtained in (c) above on a Venn diagram (4 Marks)

d) Define:

i) A mathematical function (2Marks)

ii) A mathematical relation

**QUESTION TWO**

a) Define the following two methods of set enumeration

i) Roster Method

ii) Set builder Method (4Marks)

b) Define the prime numbers from 7 𝑡𝑜 16 using the above two methods of set enumeration

(4Marks)

c) With the help of a truth table, show that (i) and (ii) below are a contradiction and a

tautology respectively

i) (p ∨ q ) ∧ [( ̴ p ) ∧ ( ̴ q) ] (4Marks)

ii) [(p → q) ∧ p] → q (4Marks)

d) Define, with examples, the following types of sets

i) Finite Set

ii) Proper Subset

iii) Disjoint Set

iv) Universal Set (4Marks)

**QUESTION THREE**

a) Determine the hypothesis and consequences for the following conditional Statements. Then

determine their truth values.

i) The moon is square only if the sun rises in the east (3 Marks)

ii) “If you do your homework, you will not be punished.” (3 Marks)

b) In order to determine game preferences in a certain institution with a population of 700

students, a census was conducted and the following information was collected:

Those that played football – 550

Those that played rugby – 300

Those that played neither of the two games – 50

Required:

i) The number of students who played both games (3 Marks)

ii) The number of students who played just one of the two games (3 Marks)

c) Define, as used in mathematics

i) Functions, (2 Marks)

ii) Relations, (2 Marks)

iii) Domain, (2 Marks)

iv) Range. (2 Marks)

**QUESTION FOUR**

a) Find the domains and ranges of the following functions

**QUESTION FIVE**

a) Differentiate between the following terms and state an example for each

i) Propositional connectives

ii) Compound propositions (4Marks)

b) Given the following statement “If you study hard, then you will not fail in ” Foundations of

Mathematics”. Write the:

i) Negation

ii) Contra positive

iii) Converse

iv) Inverse of the statement. (4Marks)

c) Given the two sets: A = {a, b} and B = {1,2},

i) Find the Cartesian product of A and B.

ii) The Cartesian product of B and A (4Marks)

d) Differentiate between universal quantification and existential quantification (4Marks)

e) Express the statement “there is a number x such that when it is added to any number, the

result is that number, and if it is multiplied by any number, the result is x” as a logical

expression. (4Marks)