Sets and Venn Diagrams

Meaning of a Set

A set is a well-defined collection of distinct objects, called elements or members, usually represented with capital letters, e.g. A = {1, 2, 3, 4}.

Ways of Describing a Set

• Listing method – writing out all the elements, e.g. A = {2, 4, 6, 8}
• Set-builder notation – describing a rule for membership, e.g. A = {x : x is an even number less than 10}

Types of Sets

• Finite set – has a countable, limited number of elements
• Infinite set – has an uncountable or unlimited number of elements
• Empty (null) set – has no elements, written as { } or ∅
• Universal set (μ) – the set containing all elements under consideration
• Subset – a set whose elements are all contained in another set

Set Operations

• Union (A ∪ B) – all elements found in A, in B, or in both
• Intersection (A ∩ B) – elements found in both A and B
• Complement (A′) – elements in the universal set that are not in A

Venn Diagrams

A Venn diagram uses overlapping circles inside a rectangle (representing the universal set) to show the relationship between sets. Overlapping regions represent the intersection of sets, while the combined shaded area represents their union.

Worked Example

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∪ B = {1, 2, 3, 4, 5, 6} and A ∩ B = {3, 4}.

Summary

Sets provide a foundation for organising objects into groups, and Venn diagrams give a clear visual way to represent relationships such as union, intersection and complement between sets.