Sets: Definition, Types, and Operations

A set is a collection of objects or numbers with a clear boundary, so that anyone can decide with certainty whether something belongs to it.

That requirement of a clear boundary matters. The pupils of class 7A form a set, because the list is definite. The clever pupils do not form a set, because clever is a judgement - people would disagree about who belongs.

Three Ways to Write a Set

All three say the same thing.

Members are written inside curly braces, separated by commas. The order does not matter, and a repeated member is not written twice.

Types of Set

• The empty set. It has no members at all, written as empty braces. An example is the set of odd numbers divisible by two.
• The universal set. Everything currently under discussion. If we are discussing one school's pupils, the universal set is all pupils of that school.
• A subset. A set whose every member also belongs to another set. The pupils of class 7A are a subset of all year 7 pupils.
• Finite and infinite sets. Whether the members can be counted. The set of natural numbers is infinite.

Venn Diagrams

A Venn diagram makes relationships between sets visible rather than imagined. A rectangle represents the universal set, and circles inside it represent the individual sets.

This is why the sets chapter is usually a favourite: almost every question can be solved by drawing rather than calculating.