Contents

1. Relations

1.1 Introduction

1.2 Definition

1.3 Definition of relation

1.4 Types of relations

1.5 Equivalance relations

1.6 Equivalance classes

1.7 Partition of set and equivalence classes

1.8 Partial orderings

1.9 Function

2. Divisibility in Integers

2.1 Introduction

2.2 Well ordering principle

2.3 Division algorithm

2.4 Divisibility and its properties, prime numbers

2.5 Greatest common divisor

2.6 EuclideAn algorithm

2.7 Unique factorisation in integer’s

2.8 Congruence relation and its properties

2.9 Fermat’s theorem

2.10 Residue classes

3. Boolean Algebra

3.1 Introduction

3.2 Hasse diagram

3.3 Lattice

3.4 Boolean algebra

3.5 Finite boolean algebras

3.6 Boolean expressions and boolean functions

4. Sequence of Real Numbers

4.1 Definition, notation and examples

4.2 Convergent, divergent and oscillatory sequences–definition and examples

4.3 Bounded sets, bounded sequences–definition and examples

4.4 Monotonic sequences, theorem on monotonic and bounded sequences (Without Proof)

4.5 The limit

4.6 Convergence of the sequence , x ? R, x > 0

5. Series of Real Numbers

5.1 Sequence of partial sums

5.2 Convergence of the series ?un

5.3 Convergence of a geometric series

5.4 Comparison test and its limit form (Without Proof)

5.5 Convergence of ? (P-series), p?R

5.6 D’Alembert’s ratio test

5.7 Root test (Cauchy’s nth Root Test)

6. Continuity and Mean Value Theorems

6.1 Introduction

6.2 Continuity

6.3 Differentiability

6.4 Differentiability and continuity

6.5 Rolle’s theorem

6.6 Lagrange’s mean value theorem

6.7 Cauchy’s mean value theorem

7. Successive Differentiation

7.1 Introduction

7.2 nth derivatives of standard functions

7.3 Leibnitz’s theorem

7.4 L’Hospital’s rules

8. Taylor’s and maclaurin’s theorems

8.1 Taylor’s and maclaurin’s theorems

8.2 Power series expansion