This worksheet provides five distinct scenarios for students to practice filling in Venn diagrams and calculating set intersections and differences.

Let's solve each problem step by step using Venn diagrams and set theory principles.

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Problem 1: Pizza Toppings

Given:
- Total customers = 100
- Customers who ordered mushrooms (M) = 80
- Customers who ordered pepperoni (P) = 72
- Customers who ordered both mushrooms and pepperoni = 60

#### Venn Diagram Setup:
- Let \( M \cap P \) represent the intersection (both toppings).
- \( M \setminus P \) represents mushrooms but no pepperoni.
- \( P \setminus M \) represents pepperoni but no mushrooms.
- \( M^c \cap P^c \) represents neither topping.

#### Step-by-Step Solution:
1. Both toppings (\( M \cap P \)):
\[
|M \cap P| = 60
\]

2. Mushrooms but no pepperoni (\( M \setminus P \)):
\[
|M \setminus P| = |M| - |M \cap P| = 80 - 60 = 20
\]

3. Pepperoni but no mushrooms (\( P \setminus M \)):
\[
|P \setminus M| = |P| - |M \cap P| = 72 - 60 = 12
\]

4. Neither topping (\( M^c \cap P^c \)):
\[
|M^c \cap P^c| = \text{Total} - (|M| + |P| - |M \cap P|)
\]
\[
|M^c \cap P^c| = 100 - (80 + 72 - 60) = 100 - 92 = 8
\]

#### Answers:
a) Mushrooms but no pepperoni: \( \boxed{20} \)
b) Pepperoni but no mushrooms: \( \boxed{12} \)
c) Neither topping: \( \boxed{8} \)

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Problem 2: Car Repairs

Given:
- Total cars inspected = 50
- Cars needing new brakes (B) = 23
- Cars needing a new exhaust system (E) = 34
- Cars needing neither repair = 6

#### Venn Diagram Setup:
- Let \( B \cap E \) represent the intersection (both repairs).
- \( B \setminus E \) represents new brakes but no exhaust.
- \( E \setminus B \) represents new exhaust but no brakes.
- \( B^c \cap E^c \) represents neither repair.

#### Step-by-Step Solution:
1. Cars needing either or both repairs:
\[
\text{Cars needing either or both} = \text{Total} - \text{Neither} = 50 - 6 = 44
\]

2. Both repairs (\( B \cap E \)):
Using the principle of inclusion-exclusion:
\[
|B \cup E| = |B| + |E| - |B \cap E|
\]
\[
44 = 23 + 34 - |B \cap E|
\]
\[
|B \cap E| = 23 + 34 - 44 = 13
\]

3. New brakes but not exhaust (\( B \setminus E \)):
\[
|B \setminus E| = |B| - |B \cap E| = 23 - 13 = 10
\]

#### Answers:
a) Both repairs: \( \boxed{13} \)
b) New brakes but not exhaust: \( \boxed{10} \)

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Problem 3: Subject Preferences

Given:
- Total students = 85
- Students liking math (M) = 35
- Students liking history (H) = 37
- Students liking both math and history = 20

#### Venn Diagram Setup:
- Let \( M \cap H \) represent the intersection (both subjects).
- \( M \setminus H \) represents math but not history.
- \( H \setminus M \) represents history but not math.
- \( M^c \cap H^c \) represents neither subject.

#### Step-by-Step Solution:
1. Math but not history (\( M \setminus H \)):
\[
|M \setminus H| = |M| - |M \cap H| = 35 - 20 = 15
\]

2. History but not math (\( H \setminus M \)):
\[
|H \setminus M| = |H| - |M \cap H| = 37 - 20 = 17
\]

3. Neither subject (\( M^c \cap H^c \)):
\[
|M^c \cap H^c| = \text{Total} - (|M| + |H| - |M \cap H|)
\]
\[
|M^c \cap H^c| = 85 - (35 + 37 - 20) = 85 - 52 = 33
\]

4. Math or history (\( M \cup H \)):
\[
|M \cup H| = |M| + |H| - |M \cap H| = 35 + 37 - 20 = 52
\]

#### Answers:
a) Math or history: \( \boxed{52} \)
b) Neither of the two subjects: \( \boxed{33} \)
c) Math but not history: \( \boxed{15} \)

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Problem 4: Sports Participation

Given:
- Total students = 80
- Students playing basketball (B) = 56
- Students playing volleyball (V) = 30
- Students playing both sports = 14

#### Venn Diagram Setup:
- Let \( B \cap V \) represent the intersection (both sports).
- \( B \setminus V \) represents basketball but not volleyball.
- \( V \setminus B \) represents volleyball but not basketball.
- \( B^c \cap V^c \) represents neither sport.

#### Step-by-Step Solution:
1. Basketball but not volleyball (\( B \setminus V \)):
\[
|B \setminus V| = |B| - |B \cap V| = 56 - 14 = 42
\]

2. Volleyball but not basketball (\( V \setminus B \)):
\[
|V \setminus B| = |V| - |B \cap V| = 30 - 14 = 16
\]

3. Neither sport (\( B^c \cap V^c \)):
\[
|B^c \cap V^c| = \text{Total} - (|B| + |V| - |B \cap V|)
\]
\[
|B^c \cap V^c| = 80 - (56 + 30 - 14) = 80 - 72 = 8
\]

#### Answers:
a) Didn't play either sport: \( \boxed{8} \)
b) Played volleyball only: \( \boxed{16} \)
c) Played basketball only: \( \boxed{42} \)

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Problem 5: Holiday Activities

Given:
- Total people = 24
- People going swimming (S) = 15
- People going fishing (F) = 12
- People doing neither = 6

#### Venn Diagram Setup:
- Let \( S \cap F \) represent the intersection (both activities).
- \( S \setminus F \) represents swimming but not fishing.
- \( F \setminus S \) represents fishing but not swimming.
- \( S^c \cap F^c \) represents neither activity.

#### Step-by-Step Solution:
1. People doing either or both activities:
\[
\text{People doing either or both} = \text{Total} - \text{Neither} = 24 - 6 = 18
\]

2. Both activities (\( S \cap F \)):
Using the principle of inclusion-exclusion:
\[
|S \cup F| = |S| + |F| - |S \cap F|
\]
\[
18 = 15 + 12 - |S \cap F|
\]
\[
|S \cap F| = 15 + 12 - 18 = 9
\]

3. Swimming but not fishing (\( S \setminus F \)):
\[
|S \setminus F| = |S| - |S \cap F| = 15 - 9 = 6
\]

4. Fishing but not swimming (\( F \setminus S \)):
\[
|F \setminus S| = |F| - |S \cap F| = 12 - 9 = 3
\]

#### Answers:
a) Go swimming and fishing: \( \boxed{9} \)
b) Only went swimming: \( \boxed{6} \)
c) Only went fishing: \( \boxed{3} \)

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Final Answers:

1. a) \( \boxed{20} \), b) \( \boxed{12} \), c) \( \boxed{8} \)
2. a) \( \boxed{13} \), b) \( \boxed{10} \)
3. a) \( \boxed{52} \), b) \( \boxed{33} \), c) \( \boxed{15} \)
4. a) \( \boxed{8} \), b) \( \boxed{16} \), c) \( \boxed{42} \)
5. a) \( \boxed{9} \), b) \( \boxed{6} \), c) \( \boxed{3} \)