10.2 Diagrams and Set Notation
Understanding Sets and Venn Diagrams
A set is a collection of objects. Venn diagrams use circles to show relationships between sets visually.
A set is a collection of objects. Venn diagrams use circles to show relationships between sets visually.
⚡ Set Notation:
Universal Set (ξ or U)
All elements under consideration
Element (∈)
$5 \in A$ means "5 is in set A"
Not Element (∉)
$7 \notin A$ means "7 is not in A"
Empty Set (∅)
Set with no elements
Subset (⊆)
$A \subseteq B$ means A is contained in B
Number of Elements (n)
$n(A) = 5$ means set A has 5 elements
⚡ Set Operations:
Union (∪)
A ∪ B
Elements in A OR B (or both)
Intersection (∩)
A ∩ B
Elements in BOTH A AND B
Complement (A')
A'
Elements NOT in A
Example 1: Basic Venn Diagram
In a class of 30 students:
• 18 study French (F)
• 15 study Spanish (S)
• 8 study both languages
Draw a Venn diagram and find how many study neither language.
Solution:
Step 1: Start with intersection (both)
Both languages = 8
Step 2: Find only French
Total French - Both = 18 - 8 = 10
Step 3: Find only Spanish
Total Spanish - Both = 15 - 8 = 7
Step 4: Find neither
Total - (Only F + Both + Only S)
= 30 - (10 + 8 + 7) = 30 - 25 = 5
Answer: 5 students study neither language
• 18 study French (F)
• 15 study Spanish (S)
• 8 study both languages
Draw a Venn diagram and find how many study neither language.
Step 1: Start with intersection (both)
Both languages = 8
Step 2: Find only French
Total French - Both = 18 - 8 = 10
Step 3: Find only Spanish
Total Spanish - Both = 15 - 8 = 7
Step 4: Find neither
Total - (Only F + Both + Only S)
= 30 - (10 + 8 + 7) = 30 - 25 = 5
Answer: 5 students study neither language
Example 2: Using Set Notation
Given: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 4, 6, 8, 10} (even numbers)
B = {1, 2, 3, 4, 5} (numbers ≤ 5)
Find: (a) A ∪ B, (b) A ∩ B, (c) A', (d) n(A ∪ B)
(a) A ∪ B: Elements in A OR B
A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}
(b) A ∩ B: Elements in BOTH A AND B
A ∩ B = {2, 4}
(c) A': Elements NOT in A
A' = {1, 3, 5, 7, 9}
(d) n(A ∪ B): Number of elements in A ∪ B
n(A ∪ B) = 8
A = {2, 4, 6, 8, 10} (even numbers)
B = {1, 2, 3, 4, 5} (numbers ≤ 5)
Find: (a) A ∪ B, (b) A ∩ B, (c) A', (d) n(A ∪ B)
(a) A ∪ B: Elements in A OR B
A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}
(b) A ∩ B: Elements in BOTH A AND B
A ∩ B = {2, 4}
(c) A': Elements NOT in A
A' = {1, 3, 5, 7, 9}
(d) n(A ∪ B): Number of elements in A ∪ B
n(A ∪ B) = 8
Example 3: Three-Set Problem
In a survey of 100 people about pets:
• 45 have dogs (D)
• 38 have cats (C)
• 28 have fish (F)
• 15 have dogs and cats
• 12 have dogs and fish
• 10 have cats and fish
• 5 have all three
Find how many have no pets.
Solution using inclusion-exclusion:
Start from the center (all three) and work outward:
Only D and C (not F): 15 - 5 = 10
Only D and F (not C): 12 - 5 = 7
Only C and F (not D): 10 - 5 = 5
Only D: 45 - 10 - 5 - 7 = 23
Only C: 38 - 10 - 5 - 5 = 18
Only F: 28 - 7 - 5 - 5 = 11
Total with pets: 23 + 18 + 11 + 10 + 7 + 5 + 5 = 79
No pets: 100 - 79 = 21
Answer: 21 people have no pets
• 45 have dogs (D)
• 38 have cats (C)
• 28 have fish (F)
• 15 have dogs and cats
• 12 have dogs and fish
• 10 have cats and fish
• 5 have all three
Find how many have no pets.
Solution using inclusion-exclusion:
Start from the center (all three) and work outward:
Only D and C (not F): 15 - 5 = 10
Only D and F (not C): 12 - 5 = 7
Only C and F (not D): 10 - 5 = 5
Only D: 45 - 10 - 5 - 7 = 23
Only C: 38 - 10 - 5 - 5 = 18
Only F: 28 - 7 - 5 - 5 = 11
Total with pets: 23 + 18 + 11 + 10 + 7 + 5 + 5 = 79
No pets: 100 - 79 = 21
Answer: 21 people have no pets
💡 Venn Diagram Tips:
• Always start with the intersection: The overlap comes first
• Work outward: Then fill in "only A" and "only B"
• Don't forget the outside: Elements in neither set
• Check your answer: All regions should add up to the total
• Use n(A) for counting: The notation n(A) means "number of elements in A"
• Work outward: Then fill in "only A" and "only B"
• Don't forget the outside: Elements in neither set
• Check your answer: All regions should add up to the total
• Use n(A) for counting: The notation n(A) means "number of elements in A"
🎯 Venn Diagram Practice:
Understanding Frequency Trees
Frequency trees show how a total is split into categories and subcategories. They're particularly useful for two-way data.
Frequency trees show how a total is split into categories and subcategories. They're particularly useful for two-way data.
⚡ When to Use Frequency Trees:
Perfect for:
• Sorting data into categories (Male/Female, Pass/Fail, etc.)
• Finding totals for combined characteristics
• Calculating probabilities from data
• Two-way tables converted to tree format
• Sorting data into categories (Male/Female, Pass/Fail, etc.)
• Finding totals for combined characteristics
• Calculating probabilities from data
• Two-way tables converted to tree format
Example 1: Basic Frequency Tree
A school has 200 students. 120 are boys and 80 are girls. Of the boys, 75 passed an exam. Of the girls, 55 passed.
Draw a frequency tree and find:
(a) How many students passed?
(b) What is the probability a randomly selected student is a girl who passed?
Solutions:
(a) Total passed = 75 + 55 = 130 students
(b) P(girl who passed) = $\frac{55}{200} = \frac{11}{40} = 0.275$ or 27.5%
Draw a frequency tree and find:
(a) How many students passed?
(b) What is the probability a randomly selected student is a girl who passed?
(a) Total passed = 75 + 55 = 130 students
(b) P(girl who passed) = $\frac{55}{200} = \frac{11}{40} = 0.275$ or 27.5%
Example 2: Working Backwards
A survey of 150 people asked about coffee preference:
• 90 prefer hot drinks
• 35 prefer hot coffee
• 40 prefer cold coffee
Complete the frequency tree.
Solution:
Total cold drinks = 150 - 90 = 60
Hot drinks (90):
• Hot coffee: 35
• Hot not-coffee: 90 - 35 = 55
Cold drinks (60):
• Cold coffee: 40
• Cold not-coffee: 60 - 40 = 20
Check: 35 + 55 + 40 + 20 = 150 ✓
• 90 prefer hot drinks
• 35 prefer hot coffee
• 40 prefer cold coffee
Complete the frequency tree.
Solution:
Total cold drinks = 150 - 90 = 60
Hot drinks (90):
• Hot coffee: 35
• Hot not-coffee: 90 - 35 = 55
Cold drinks (60):
• Cold coffee: 40
• Cold not-coffee: 60 - 40 = 20
Check: 35 + 55 + 40 + 20 = 150 ✓
Example 3: Finding Probabilities
100 light bulbs are tested. 60 are LED and 40 are incandescent. Of the LED bulbs, 57 work and 3 are faulty. Of the incandescent bulbs, 35 work and 5 are faulty.
Find:
(a) P(bulb works)
(b) P(LED and faulty)
(c) P(faulty | incandescent)
Solutions:
(a) Total working = 57 + 35 = 92
$P(\text{works}) = \frac{92}{100} = 0.92$ or 92%
(b) LED and faulty = 3
$P(\text{LED and faulty}) = \frac{3}{100} = 0.03$ or 3%
(c) This is conditional probability
Of 40 incandescent, 5 are faulty
$P(\text{faulty | incandescent}) = \frac{5}{40} = \frac{1}{8} = 0.125$ or 12.5%
Find:
(a) P(bulb works)
(b) P(LED and faulty)
(c) P(faulty | incandescent)
Solutions:
(a) Total working = 57 + 35 = 92
$P(\text{works}) = \frac{92}{100} = 0.92$ or 92%
(b) LED and faulty = 3
$P(\text{LED and faulty}) = \frac{3}{100} = 0.03$ or 3%
(c) This is conditional probability
Of 40 incandescent, 5 are faulty
$P(\text{faulty | incandescent}) = \frac{5}{40} = \frac{1}{8} = 0.125$ or 12.5%
Example 4: Two-Way Table to Frequency Tree
Convert this two-way table to a frequency tree:
Frequency Tree:
• Start: 100
• Split: 60 adults, 40 children
• Adults: 45 with car, 15 no car
• Children: 0 with car, 40 no car
| Own Car | No Car | Total | |
|---|---|---|---|
| Adult | 45 | 15 | 60 |
| Child | 0 | 40 | 40 |
| Total | 45 | 55 | 100 |
• Start: 100
• Split: 60 adults, 40 children
• Adults: 45 with car, 15 no car
• Children: 0 with car, 40 no car
💡 Frequency Tree Tips:
• Start at the top: Begin with the total
• Check totals: Branches should always add up
• Label clearly: Write what each branch represents
• End values: The final branches show the actual frequencies
• Check totals: Branches should always add up
• Label clearly: Write what each branch represents
• End values: The final branches show the actual frequencies
🎯Frequency Tree Practice
Real Life Uses:
• Market research: Customer preferences and demographics
• Medical studies: Treatment outcomes by patient group
• Quality control: Defect rates by production line
• Education: Test results by class or year
• Sports analysis: Performance statistics by category
• Opinion polls: Responses by age group or region
• Market research: Customer preferences and demographics
• Medical studies: Treatment outcomes by patient group
• Quality control: Defect rates by production line
• Education: Test results by class or year
• Sports analysis: Performance statistics by category
• Opinion polls: Responses by age group or region