SETS


Topic 1   The development of Set Theory

Learning objective:  The language of sets

•  Set definitions and operations
•  Well-defined sets, equal sets and set difference
•  Partitions and Venn diagram
•  Venn diagram and Set properties
•  The Cartesian product of two sets
•  Relations
•  Equivalence relations, Modular Congruence
•  Equivalence classes and partitions

 


 
 
Topic 2   Extension of concepts of functions and introduction

Learning objective:  Evolution of function concept

•  Functions as relations
•  Equality of functions
•  Composite of functions
•  Inverse functions
•  Properties of functions
•  Identity functions
•  Binary operations
•  Properties of binary operations
•  The identity element  e
•  The inverse of an element
•  The cancellation laws

 
 


 
 Topic 3   The Universal Theory of Everything in Mathematics

Learning objective:  Group Theory

•  Groups
• Infinite groups
• Finite groups
• Groups of integers modulo n
• Symmetry groups
• Properties and theorems of groups and subgroups
• Right and left cancellation laws of groups
• Subgroups
• Cyclic groups

 
 



Topic 4   The classifications of groups

Learning objective:  Group structures

• Permutation groups
• Permutation and cycle form
• Properties of cycle form
• Cosets and Lagrange’s theorem
• Homomorphisms
• The kernel of a homomorphisms
• Isomorphisms​