Mr R-C's Classes
  • Home
  • IGCSE
    • IGCSE Unit Resources >
      • Unit 0 - Simultaneous
      • Unit 0 - Regression
      • Unit 1 - Percentages
      • Unit 2 - Sets
      • Unit 3 - Indices and Surds
      • Unit 4 - Circle Theorems
      • Unit 5 - Formulae
      • Unit 6 - Vectors
      • Unit 7 - Quadratic Equations
      • Unit 8 - Tree Diagrams
      • Unit 9 - Coordinate Geometry
      • Unit 10 - Sequences
      • Unit 11 - Statistical Diagrams
      • Unit 12 - Composite and Inverse Functions
      • Unit 13 - Similarity
      • Unit 14 - Variation
      • Unit 15 - Algebraic Fractions
      • Unit 16 - Advanced Trigonometry
      • Unit 17 - Quadratic Functions
      • Unit 18 - Volumes and Surface Area
      • Unit 19 - Graphs
      • Unit 20 - Logarithms
      • Unit 21 - Transforming Graphs
    • IGCSE Revision Materials
  • IB
    • Mathematical Studies >
      • MSSL Unit Resources >
        • Unit 1 - Number Review
        • Unit 2 - Sequences and Series
        • Unit 3 - Statistics 1
        • Unit 4 - Functions 1
        • Unit 5 - Trigonometry and Mensurations
        • Unit 6 - Functions 2
        • Unit 7 - Statistics 2
        • Unit 8 - Differentiation
        • Unit 9 - Sets and Probability
        • Unit 10 - Functions 3
        • Unit 11 - Financial Maths
        • Unit 12 - Statistics 3
        • Unit 13 - Logic
      • IB Studies Revision Homework
      • MSSL Revision Materials
      • Studies Exam Question Practice
    • Mathematics Higher Level >
      • HL Unit Resources >
        • Unit 1 - Algebra
        • Unit 2 - Functions & Equations
        • Unit 3 - Trigonometry
        • Unit 4 - Exponentials and Logarithms
        • Unit 5 - Statistics and Probability
        • Unit 6 - Differential Calculus
        • Unit 7 - Discrete Probability Distributions
        • Unit 8 - Integral Calculus
        • Unit 9 - Continuous Probability Distributions
        • Unit 10 - Complex Numbers
        • Unit 11A - Statistics Option
        • Unit 11B - Sets and Groups Option
        • Unit 12 - Vectors
      • HL Revision Materials >
        • HL Weekly Quizzes
    • Mathematics AA SL >
      • AASL Unit Resources >
        • Unit 0 - Proof
        • Unit 1 - Algebra Functions Basics
        • Unit 2 - Functions, Equations and Inequalities
        • Unit 3 - Sequences and Series
        • Unit 4 - Exponentials and Logarithms
        • Unit 5 - Trigonometric Functions and Equations
        • Unit 6 - Differentiation 1
        • Unit 7 - Statistics
      • AASL Revision Materials
      • AASL Key Skills Generator

Unit 11 - Sets, Relations and Groups Option

Unit Objectives
By the end of this unit students should be able to:
  • Work with finite and infinite sets, including subsets and proving sets are equal
  • Perform calculations with binary operations, and visualise binary operations on a set in a Cayler Table
  • Show that a binary operation on a set is: closed; associative; distributive; commutative.
  • Find the identity element (if it exists) of a binary operation on a set
  • Find inverse elements (if they exist) of a binary operation on a set
  • Prove and use the Left and Right Cancellation Laws
  • Prove and use the uniqueness of identity elements and inverses
  • Find the intersection, union, complement, set difference and symmetric difference when working with sets
  • Prove and use: De Morgan's Laws; distributivity, associativity and commutativity of set union and set intersection
  • Work with simple modular arithmetic
  • Find the Cartesian product of two sets
  • Identify relations, and decide if they are reflexive, symmetric or transitive
  • Determine if a relation is an equivalence relation and find the equivalence classes
  • Identify if a relation is a function, and work out the domain, codomain and range of a function
  • Determine if a function is an injection, surjection or bijection
  • Compose functions and find inverse functions
  • Understand that inverse functions only exist if it is a bijection, and that composition is not commutative
  • Prove that a given set with a binary operation is (or is not) a group, and determine if it is Abelian
  • Recognise the common groups: Z, Q, R, C under addition; integers modulo n under addition; non-zero integers modulo p under multiplication; symmetries of plane figures; bijective functions
  • Find the order of a group and the elements of a group
  • Determine if a group is cyclic and find its generators
  • Prove that all cyclic groups are Abelian
  • Compose permutations written in both normal and cycle form
  • Use the fact that all permutations can be written as a composition of disjoint cycles
  • Find the order of a permutation in cycle form
  • Prove that a subset of a group forms a subgroup
  • Prove and use the subgroup tests for infinite and finite groups
  • Find the (left and right) cosets of a subgroup of a group
  • Use Lagrange's Theorem to determine the order of possible subgroups
  • Prove and use the Corollary to Lagrange's Theorem that the order of any element is a factor of the order of the group
  • Determine if a function between two groups is a homomorphism
  • Find the kernel and range of a homomorphism
  • Prove that the kernel and range of a homomorphism are subgroups of the domain and codomain respectively
  • Prove and use the result that identities and inverses map to each other under a homomorphism
  • Determine if two groups are isomorphic, by finding an isomorphism between them
  • Use the fact that the order of an element is unchanged by an isomorphism
Lesson Resources​
0. Unit Map and Glossary
Unit Map
Glossary
1. Further Proof
Lesson Sheet
2. Sets
Lesson Sheet
3. Binary Operations
Lesson Sheet
4. Operations on Sets
Lesson Sheet
Summary Table
5. Congruence
Lesson Sheet
6. Relations
Lesson Sheet
7. Equivalence Relations
Lesson Sheet
8. Functions
Lesson Sheet
9. Composition of Functions
Lesson Sheet
10. Groups
Lesson Sheet
Lesson Sheet COMPLETE
Rings and Fields
Rings and Fields COMPLETE
11. Order
Lesson Sheet
Lesson Sheet COMPLETE
12. Cyclic Groups
Lesson Sheet
Lesson Sheet COMPLETE
13. Subgroups
Lesson Sheet
Lesson Sheet COMPLETE
14. Cosets
Lesson Sheet
Lesson Sheet COMPLETE
15. Lagrange's Theorem
Lesson Sheet
Lesson Sheet COMPLETE
16. Common Finite Groups
Lesson Sheet
Lesson Sheet COMPLETE
17. Permutations
Lesson Sheet
Lesson Sheet COMPLETE
18. Symmetries of Shapes
Lesson Sheet
Lesson Sheet COMPLETE
19. Homomorphisms and Kernels
Lesson Sheet
Lesson Sheet COMPLETE
20. Isomorphisms
Lesson Sheet
Lesson Sheet COMPLETE
For a whole range of mathematical activities for use in and out of the classroom, including random question generators, Bingo generators and worksheet generators, as well as computer based investigations and activities check out my main site: www.interactive-maths.com.

​If you are interested in codes, ciphers and codebreaking, then check out my secondary site, Crypto Corner, dedicated to providing information about all types of ciphers, as well as tools for encrypting, decrypting and breaking messages.
  • Home
  • IGCSE
    • IGCSE Unit Resources >
      • Unit 0 - Simultaneous
      • Unit 0 - Regression
      • Unit 1 - Percentages
      • Unit 2 - Sets
      • Unit 3 - Indices and Surds
      • Unit 4 - Circle Theorems
      • Unit 5 - Formulae
      • Unit 6 - Vectors
      • Unit 7 - Quadratic Equations
      • Unit 8 - Tree Diagrams
      • Unit 9 - Coordinate Geometry
      • Unit 10 - Sequences
      • Unit 11 - Statistical Diagrams
      • Unit 12 - Composite and Inverse Functions
      • Unit 13 - Similarity
      • Unit 14 - Variation
      • Unit 15 - Algebraic Fractions
      • Unit 16 - Advanced Trigonometry
      • Unit 17 - Quadratic Functions
      • Unit 18 - Volumes and Surface Area
      • Unit 19 - Graphs
      • Unit 20 - Logarithms
      • Unit 21 - Transforming Graphs
    • IGCSE Revision Materials
  • IB
    • Mathematical Studies >
      • MSSL Unit Resources >
        • Unit 1 - Number Review
        • Unit 2 - Sequences and Series
        • Unit 3 - Statistics 1
        • Unit 4 - Functions 1
        • Unit 5 - Trigonometry and Mensurations
        • Unit 6 - Functions 2
        • Unit 7 - Statistics 2
        • Unit 8 - Differentiation
        • Unit 9 - Sets and Probability
        • Unit 10 - Functions 3
        • Unit 11 - Financial Maths
        • Unit 12 - Statistics 3
        • Unit 13 - Logic
      • IB Studies Revision Homework
      • MSSL Revision Materials
      • Studies Exam Question Practice
    • Mathematics Higher Level >
      • HL Unit Resources >
        • Unit 1 - Algebra
        • Unit 2 - Functions & Equations
        • Unit 3 - Trigonometry
        • Unit 4 - Exponentials and Logarithms
        • Unit 5 - Statistics and Probability
        • Unit 6 - Differential Calculus
        • Unit 7 - Discrete Probability Distributions
        • Unit 8 - Integral Calculus
        • Unit 9 - Continuous Probability Distributions
        • Unit 10 - Complex Numbers
        • Unit 11A - Statistics Option
        • Unit 11B - Sets and Groups Option
        • Unit 12 - Vectors
      • HL Revision Materials >
        • HL Weekly Quizzes
    • Mathematics AA SL >
      • AASL Unit Resources >
        • Unit 0 - Proof
        • Unit 1 - Algebra Functions Basics
        • Unit 2 - Functions, Equations and Inequalities
        • Unit 3 - Sequences and Series
        • Unit 4 - Exponentials and Logarithms
        • Unit 5 - Trigonometric Functions and Equations
        • Unit 6 - Differentiation 1
        • Unit 7 - Statistics
      • AASL Revision Materials
      • AASL Key Skills Generator