This course builds the language of sets and relations that later mathematics is written in. It starts with what a set is, roster and set-builder form, and the standard number sets, then compares sets through cardinal number, subsets and the power set. Venn diagrams show union, intersection, difference, symmetric difference and complement, and the laws of the algebra of sets, including De Morgan's laws, simplify set expressions. Counting results for two and three sets solve survey problems. The course ends with ordered pairs, the Cartesian product, relations and their domain, range and inverse, and reflexive, symmetric, transitive and equivalence relations.
Middle Years
Sets and Relations
Sets and how to write them, subsets and power sets, Venn diagrams and set operations, the laws of sets, counting with cardinal numbers, survey problems, and relations with their domain, range, inverse and types.
- 10 lessons
Overview
Lessons
-
Sets and how to write them
What a set is, roster form and set-builder form, the empty, singleton, finite and infinite sets, and the standard number sets N, W, Z, Q, R and C.
-
Cardinal number, subsets and the power set
Cardinal number, equal and equivalent sets, subsets and proper subsets, why a set with n elements has 2ⁿ subsets, the power set and the universal set.
-
Venn diagrams and operations on sets
Venn diagrams and the operations of union, intersection, difference, symmetric difference and complement, with disjoint sets and worked examples.
-
Laws of the algebra of sets
Idempotent, identity, commutative, associative and distributive laws, De Morgan's laws, and using the laws to simplify set expressions.
-
Counting elements: cardinal number results
Formulas for the number of elements in unions, differences, complements and symmetric differences, the largest and smallest possible counts, and results for three sets.
-
Problems on sets
Sets of multiples such as 3N ∩ 7N, coprime numbers, sets defined by a formula, counting subsets, and simplifying set expressions.
-
Word problems with Venn diagrams
Translating phrases such as at least one, both, only and neither into sets, then solving survey problems with two and three sets region by region.
-
Cartesian product and relations
Ordered pairs, the Cartesian product A × B and its size, relations as subsets of A × B, the number of relations, and the void and universal relations.
-
Domain, range and inverse of a relation
The domain, range and codomain of a relation, arrow diagrams, and the inverse relation obtained by reversing every pair.
-
Types of relations
Reflexive, symmetric, transitive and antisymmetric relations, equivalence relations and partial orders, tested on finite sets and on relations given by a rule.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Sets and Relations
Typed from the teacher's handwritten class slides, 4 pages.
Download the notes (PDF, 135.5 KB)