Contents

Preface

Acknowledgements

Author Biography

1 Equality ‘=’

2 Fundamental Properties of Sets

2.1 What is a Set?

2.2 Defining a Set

2.3 Equality of Sets

2.4 A Set Can Be an Element of Another Set

2.5 A Set Cannot Contain Itself

2.6 A Set Can Be Empty (Null )

2.7 Subsets

2.7.1 Definition of Subsets

2.7.2 A ⊂ A

2.7.3 ∅ ⊂ A

3 Set Operators

3.1 What is a set operator?

3.2 The ∪ Operator (UNION)

3.2.1 The Commutative Property of ∪

3.2.2 The Associative Property of ∪

3.2.3 The ∪ operator and the empty set ∅

3.3 The ∩ Operator (INTERSECTION)

3.3.1 The Commutative Property of ∩

3.3.2 The Associative Property of ∩

3.3.3 The ∩ operator and the empty set ∅

3.4 Mixed Properties of ∪ and ∩

3.5 The \ Operator (SET SUBSTRACTION)

3.5.1 The \ operator and the empty set ∅

3.6 Mixed Properties of \ , ∪ and ∩

3.7 The × Operator (CARTESIAN PRODUCT)

3.7.1 The × operator and the empty set ∅

4 Universal Set Systems

4.1 What is a Universal Set?

4.2 Complement

4.3 Properties of the Complement

5 Relations and Functions

5.1 What is a Relation?

5.2 One-to-one Relations

5.3 What is a Function?

6 Equivalence Relations and Classes

6.1 What is an Equivalence Relation?

6.2 What is an Equivalence Class?

7 Mathematical Theory

7.1 Axiomatic Definitions and Definitions

7.2 Axioms and Theorems

8 Appendix

8.1 General Equations of Set Theory

8.2 Universal Set Systems

8.3 Relations and Functions

8.3.1 Relations

8.3.2 One-to-one Relations

8.3.3 Functions

8.4 Equivalence Relations and Classes

8.4.1 Equivalence Relations

8.4.2 Equivalence Classes