← Unit 4 Venn Diagrams and Set Notation

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MA5-PRO-C-01

Venn Diagrams and Set Notation

⏱ 25 min📚 Year 9📈
Think First

In a class of 30, 18 play tennis, 15 play basketball, and 8 play both. How many play neither?

💡 Revision: Ensure you understand basic probability and can work with fractions.

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
UnionIntersectionMutually exclusiveInclusion-exclusion
Learn Phase
1

Set Notation

The language of probability

Key notation:

  • $A cup B$: unionA or B or both
  • $A cap B$: intersectionA and B
  • $A'$: complementnot 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}

2

Venn Diagrams

Visualising overlaps

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

3

The Addition Rule

P(A or B)

For any two events:

Addition Rule

$P(A cup B) = P(A) + P(B) - P(A cap B)$

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

Try it yourself

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.

Worked Example

Venn Diagrams

1

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

2

P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A ∪ B).

$P(A cup B) = 0.4 + 0.5 - 0.2 = 0.7$

3

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).

Your Turn

Practice, Venn Diagrams

Work through each question in your book or digitally. Answers are in the Questions phase.

1Draw a Venn diagram: 40 students, 25 play sport, 20 play music, 12 do both. Find how many do neither.
2P(A)=0.3, P(B)=0.4, P(A∩B)=0.1. Find P(A∪B) and P(A∩B').
3In a deck, find P(red or face card).
Real-World Anchor

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)
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
$A cap B$ means:
If A and B are mutually exclusive, $P(A cap B) = $
P(A)=0.3, P(B)=0.4, P(A∩B)=0.1. P(A∪B)=
In Venn diagram, overlap shows:
30 students, 20 like Maths, 15 like Science, 10 like both. Neither:
P(heart or king) from deck:
$A cup B$ in words:
If P(A∪B)=0.8, P(A)=0.5, P(B)=0.4, then P(A∩B)=
1In a class of 35, 20 study French, 18 study German, 12 study both. Draw a Venn diagram and find how many study neither.
2P(A)=0.6, P(B)=0.5, P(A∩B)=0.3. Find P(A∪B) and P(neither).
3Explain why the addition rule requires subtracting P(A∩B).

Comprehensive Answers

135 students, 20 French, 18 German, 12 both.
French only=8, German only=6, Both=12, Neither=9.
2P(A)=0.6, P(B)=0.5, P(A∩B)=0.3.
P(A∪B)=0.6+0.5-0.3=0.8. P(neither)=1-0.8=0.2.
3Why subtract P(A∩B).
The overlap is counted twice when adding P(A)+P(B). Subtract once to correct.
MC 1$A cap B$ meaning.
A and B. Answer: B
MC 2Mutually exclusive P(A∩B).
0. Answer: C
MC 3P(A∪B) with given values.
0.3+0.4-0.1=0.6. Answer: A
MC 4Overlap in Venn.
Intersection. Answer: B
MC 530 students, neither.
30-(10+5+10)=5. Answer: B
MC 6P(heart or king).
16/52=4/13. Answer: D
MC 7$A cup B$ in words.
A or B or both. Answer: B
MC 8Find P(A∩B).
0.5+0.4-0.8=0.1. Answer: A
SA 1Venn diagram for languages.
Neither=9.
SA 2P(A∪B) and P(neither).
0.8 and 0.2.
SA 3Why subtract overlap.
Avoid double-counting.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.
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