Venn diagrams are useful for showing sets, and situations involving probability.
Venn diagrams usually consist of a rectangle which shows the univeral set, U or total sample space, and circles which show sets or events.
Set Symbols
To refresh memories about the set notation, this table is copied from an earlier topic.
Symbol

Meaning

Example

U or ξ

the universal set  the set of all elements or members being considered. 
φ or {}

the empty or null set  This set has no elements or members. 
is a member of  2 {even numbers}  
is not a member of  3 {even numbers}  
is a subset of  {3, 4} {2, 3, 4, 5}  
n(A)

the cardinal number (number of members) of set A. 
if A = {4, 5, 6, 7, 8} 
A'

the complement of A. All the members not in A 
If A = {4, 5, 6, 7, 8} and the universal set is {whole numbers less than 10 then A' = {0, 1, 2, 3, 9} 
the union of sets 
If A = {4, 5, 6, 7, 8} and B = {3, 4, 5} then 

∩

the intersection of sets  If A = {4, 5, 6, 7, 8} and B = {3, 4, 5} then A ∩ B = {4, 5} 
Venn Diagrams with Two Sets
Venn diagrams can contain two sets and the following arrangements are some of those possible.
Union of two sets

Intersection of two sets

Disjoint sets

Complement of a set

Example
The Venn diagram below shows the universal set 
Find:

Answers:

a. B ' b. A c. d. 
a. B ' = {1, 2, 3, 4, 5, 8} b. = {3, 6, 7, 8, 9, 10} c. = {10} d. = {6, 7, 9} 
Venn Diagrams with Three Sets
Venn diagrams can contain three or more sets.
The Venn diagram below shows the sports played on a weekend by 150 boys.
From the diagram it can be seen that:
60 boys play rugby  (42 + 9 + 3 + 6)  {rugby} 
5 boys play both soccer andhockey  (3 + 2)  {soccer}{hockey} 
3 boys play all three sports  3  {rugby}{soccer}{hockey} 
96 boys play either soccer or rugby 
(42+9+3+6)+(34+9+3+2) − (9+3)  {rugby}{soccer} − {rugby}{soccer} 
Note that thirty three boys play no sport at the weekend.