Outline of IB Mathematics HL option Sets, relations and groups

Here you can find a description of the course not so detailed.

You can find the official syllabus of IB maths SL on the following link of IBO

http://www.ibo.org or http://store.ibo.org

Finite and infinite sets. Subsets.Operations on sets: union, intersection; complement, set difference, symmetric difference.

De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).

Ordered pairs: the Cartesian product of two sets. Relations: equivalence relations; equivalence classes.

Functions: injections; surjections; bijections. Composition of functions and inverse functions. Binary operations.Operation tables (Cayley tables).

Binary operations: associative, distributive and commutative properties. The identity element e.

The inverse of an element a.

Proof that left-cancellation and rightcancellation by an element a hold, provided that a has an inverse.

Proofs of the uniqueness of the identity and inverse elements.

The definition of a group {G,∗}. The operation table of a group is a Latin square, but the converse is false. Abelian groups.

Examples of groups: integers under addition modulo n, non-zero integers under multiplication, modulo p, where p is prime,

symmetries of plane figures, including, equilateral triangles and rectangles, invertible functions under composition of functions.

The order of a group. The order of a group element. Cyclic groups. Generators. Proof that all cyclic groups are Abelian.

Permutations under composition of permutations. Cycle notation for permutations. Result that every permutation can be written as

a composition of disjoint cycles. The order of a combination of cycles. Subgroups, proper subgroups. Use and proof of subgroup tests.

Definition and examples of left and right cosets of a subgroup of a group.

Lagrange’s theorem. Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)

Definition of a group homomorphism. Definition of the kernel of a homomorphism. Proof that the kernel and range of a homomorphism are subgroups.