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
1. Further Proof
3. Binary Operations
4. Operations on Sets
5. Congruence
6. Relations
7. Equivalence Relations
8. Functions
9. Composition of Functions
19. Homomorphisms and Kernels
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