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Venn Diagrams and Set Notation
In a class of 30, 18 play tennis, 15 play basketball, and 8 play both. How many play neither?
Learning Intentions
Know
- Union
- Intersection
- Complement
- Venn diagram
Understand
- How Venn diagrams visualise overlapping events
- The inclusion-exclusion principle
Can Do
- Draw Venn diagrams from data
- Calculate P(A ∪ B) and P(A ∩ B)
- Use the addition rule
Set Notation
Key notation:
- $A cup B$: union — A or B or both
- $A cap B$: intersection — A and B
- $A'$: complement — not A
- $emptyset$: empty set (no overlap)
Example: If A = {1, 2, 3} and B = {3, 4, 5}:
$A cup B$ = {1, 2, 3, 4, 5}, $A cap B$ = {3}
Venn Diagrams
A Venn diagram uses overlapping circles to show relationships between sets.
The overlapping region represents $A cap B$.
The total area covered by both circles represents $A cup B$.
Example: In a group of 40 students, 25 study Maths, 20 study Science, 10 study both.
Only Maths: 15, Only Science: 10, Both: 10, Neither: 5
The Addition Rule
For any two events:
$P(A cup B) = P(A) + P(B) - P(A cap B)$
If A and B are mutually exclusive (cannot both occur):
$P(A cup B) = P(A) + P(B)$
Example: P(rolling even) = 1/2, P(rolling > 4) = 1/3, P(even and > 4) = 1/6 (rolling 6)
P(even or > 4) = 1/2 + 1/3 - 1/6 = 2/3
Check Understanding
In a class, 20 students have a dog, 15 have a cat, 10 have both. Find P(dog or cat) if there are 30 students.
Venn Diagrams
In a group of 50 people, 30 like tea, 25 like coffee, 15 like both. How many like neither?
Tea only: 15, Coffee only: 10, Both: 15
Neither = 50 - (15 + 10 + 15) = 10
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A ∪ B).
$P(A cup B) = 0.4 + 0.5 - 0.2 = 0.7$
A card is drawn. Find P(heart or king).
P(heart) = 13/52, P(king) = 4/52, P(heart ∩ king) = 1/52
P(heart ∪ king) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
Common Misconceptions
P(A or B) = P(A) + P(B) always. No — only for mutually exclusive events. Otherwise subtract the overlap.
The intersection is the total of both circles. No — the intersection is the overlap. The union is the total covered.
If A and B are mutually exclusive, they can still both occur. No — mutually exclusive means they cannot both occur (intersection is empty).
Practice — Venn Diagrams
Epidemiology and Public Health
Venn diagrams are used in public health to track disease co-occurrence. For example, researchers might study the overlap between diabetes and heart disease in a population. The Australian Institute of Health and Welfare uses these visualisations to show how health conditions intersect in the population.
📓 Copy Into Your Books
▼Notation
- $A cup B$ = A or B
- $A cap B$ = A and B
- $A'$ = not A
Venn diagram
- Overlap = intersection
- Total covered = union
- Outside = neither
Addition rule
- P(A∪B) = P(A) + P(B) − P(A∩B)
- For mutually exclusive: P(A∪B) = P(A) + P(B)