Mathematics 0N1
Exercises 1
1. Write the following sets in list form.
(i) The set of letters in the word banana.
(ii) cfw_ : 2 + 3 10 = 0.
(iii) cfw_ : is an integer and 3 < < 10.
2. Write the following sets in predicate form.
(i) cfw_ , , , , .
(ii) cfw_

0N1 Mathematics Lecture 6 Propositional Logic 38
Lecture 6
Propositional Logic, Continued
Notice that p ! q and q ! p are dierent; q ! p is called
the converse of p ! q.
Example. Let p = x > 2 and q = x2 > 4. Then p ! q
is If x > 2 then x2 > 4 TRUE. But q

0N1 Mathematics Lecture 18 Roots and Coefficients 87
Lecture 18
Relations between the roots and
the coe cients of a polynomial
Consider
ax2 + bx + c = a(x
The right hand side (RHS) is
a(x2
x
)(x
)
*
Typo corrected in the following
equation
x + ) = a[x2
(

23
0N1 Mathematics Lecture 4 Set Theory
Lecture 4
Set theory
The identities in (1)-(7) of the previous lecture are called the
laws of Boolean algebra. Several of them are obvious because * obvious = evident, self-evident
of the denitions of \, [ and 0 . T

14
0N1 Mathematics Lecture 3
Lecture 3
Operations on Sets
A
A[B
A\B
B
Figure 2: Sets A and B and their intersection A \ B and union
A [ B.
Suppose A and B are sets. Then A \ B denotes the set of
all elements which belong to both A and B:
A \ B = cfw_ x :

0N1 Mathematics Arrangements for the Course
2
Arrangements for the Course
Aims of 0N1
A basic course in pure mathematical topics for members
of the foundation year.
Key ingredient: language of Mathematics, including specic use of English in Mathematics.

0N1 Mathematics Lecture 1
7
Lecture Notes
Lecture 1
Sets
A set is any collection of objects, for example, set of numbers.
The objects of a set are called the elements of the set.
A set may be specied by listing its elements. For example, cfw_1, 3, 6 denot

78
0N1 Mathematics Lecture 16 Polynomials
Lecture 16
Polynomials, continued
Division with remainder
If f (x) is a polynomial and ax + b is a linear polynomial we
can investigate whether ax + b is a factor of f (x) by long
division.
For example, if ax + b

0N1 Mathematics Lecture 10 Predicate Logic Logical equivalences 57
Lecture 10
Logical equivalences
Statements can be formed from predicates by means of a mixture of connectives and quantiers.
Examples.
(i) Let p(x, y) denote x < y and let q(y) denote y 6=

0N1 Mathematics Lecture 12 Methods of Proof 64
Lecture 12
Methods of proof, continued
II Statements of the form (8x)(p(x) ! q(x)
An example is
For all x, if x > 2 then x2 > 4.
In practice such a sentence is often expressed as
If x > 2 then x2 > 4
where th

0N1 Mathematics Lectures 1920 Mathematical Induction 95
Lectures 1920
Principle of Mathematical Induction *
Recommended reading: Richard
Hammack, Book of Proof, Chapter 10.
Let p1 , p2 , p3 , . . . be an innite sequence of statements, one
statement pn for

0N1 Mathematics Lecture 14 By contradiction
72
Lecture 14
Proof by contradiction
*
Recommended reading: Richard
Hammack, Book of Proof, Chapter 6.
Suppose we want to prove some statement q. Assume
that q is false, i.e. assume q is true. Try to deduce from

ON1 Test 5 Friday 9 November 2012
This test counts 6% of assessment for the course.
Time allocated: 11:3511:50
Name:
Student Number:
Tutorial Group:
Marking scheme: 2 marks for a complete correct answer, 1 for an incomplete correct answer, 0 for an incorr

0N1 Mathematics Exercises 5 Predicate Logic
1
Exercises 5
1. In each of the following nd values of x and y which make p(x, y) true
and nd values which make p(x, y) false. (Only one example of each is
required.)
(i) p(x, y) denotes x2 + y 2 = 2.
(ii) p(x,

ON1 Test 4 Friday 26 October 2011
This test counts 4% of assessment for the course.
Time allocated: 11:4011:50
PLEASE FILL IN CLEARLY, IN BLOCK CAPITALS:
Name:
Student Number:
Tutorial Group:
Marking scheme: 2 marks for a correct answer, 0 for an incorrec

0N1 Mathematics Assignment 4
1. Determine the truth value of each of the compound statements (a), (b)
and (c) given the following information:
(i)
(ii)
(iii)
(iv)
The statement Mr Black is taller than Mr Blue is true.
The statement Mr Green is shorter tha

0N1 Mathematics Exercises 6
1
1. Find whether the following statements are true or false where the universal
set is
Z = cfw_. . . , 3, 2, 1, 0, 1, 2, 3, . . ..
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(x)(y)(z)z < x + y
(z)(x)z < x4
(x)(y)y 2 = x
(y)(x)y 2 = x
(x)(y

0N1 Mathematics
Assignment 2
1.
(i) If A B is it necessarily true that B A?
(ii) If A B and A C is it necessarily true that B = C?
(iii) If A B and B C is it necessarily true that A C?
(iv) If A B is it necessarily true that A B?
(v) If A B is it necessar

0N1 Mathematics Exercises 8
1
1. Factorise
(i) a3 b3 .
(ii) a4 b4 .
(iii) a3 + b3 . [Hint: a3 + b3 = a3 (b)3 .]
2. Let f (x) = 2x3 3x + 1. Find
(i) f (0).
(ii) f (1).
(iii) f (1).
(iv) f ( 1 ).
2
(v) f (i).
3. In each case nd a polynomial M (x) and a rema

0N1 Mathematics Exercises 9
1
1. Find the remainder when
(i) x4 5x3 + 6x2 7 is divided by (x 1)(x 3).
(ii) x4 + x2 7 is divided by x2 4.
2. Find the constants a and b such that, when x4 ax2 + b is divided by
(x + 1)2 , the remainder is 5x 2.
3. Find the r

0N1 Mathematics Exercises 10
1
1. Prove, by induction on n, that n < 2n for every positive integer n.
2. Prove, by induction on n, that
1 + 3 + 5 + + (2n 1) = n2
for every positive integer n.
3. A sequence of numbers is dened by the properties u1 = 1 and

0N1 Mathematics Exercises 7
1
1. Which of the following statements are true?
(i) 105 is divisible by 15.
(ii) 13 | 2971.
(iii) 99 is a divisor of 3960.
(iv) 75 and 192 have no common factors.
2. Find the greatest common divisors of the following pairs.
(i

ON1 Test 1 Friday 5 October 2012
This test counts 4% of assessment for the course.
Time allowed: 5 minutes
PLEASE FILL IN CLEARLY, IN BLOCK CAPITALS:
Name:
Student Number:
Tutorial Group:
Marking scheme: 2 marks for a correct answer, 0 for an incorrect an

ON1 Test 2 Friday 12 October 2012
This test counts 4% of assessment for the course.
Time allowed: 5 minutes
PLEASE FILL IN CLEARLY, IN BLOCK CAPITALS:
Name:
Student Number:
Tutorial Group:
Marking scheme: 2 marks for a correct answer, 0 for an incorrect a

ON1 Test 3 Friday 19 October 2012
This test counts 4% of assessment for the course.
Time allowed: 10 minutes
PLEASE FILL IN CLEARLY, IN BLOCK CAPITALS:
Name:
Student Number:
Tutorial Group:
Marking scheme: 2 marks for a correct answer, 0 for an incorrect

0N1 Mathematics Lecture 8 Predicate Logic
48
Lecture 8
Predicate Logic
Many mathematical sentences involve unknowns or variables.
Examples.
(i) x > 2 (where x stands for an unknown real number).
(ii) A B (where A and B stand for unknown sets).
Such senten