Unit 11B - Sets and Groups Option

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