Set Theory Worksheet for MTH 2450, covering Venn diagrams, set operations, and problem-solving with subsets and universal sets.

Problem Analysis and Solution

The provided worksheet covers various topics in set theory, including Venn diagrams, set operations, and logical reasoning. Below is a detailed solution for each problem.

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Problem 1: Venn Diagrams

#### Task:
Draw a general Venn diagram for three sets \( A \), \( B \), and \( C \) and shade the indicated regions.

#### Solution:
1. General Venn Diagram: Draw three overlapping circles representing sets \( A \), \( B \), and \( C \). Label each region clearly.

2. Shading Regions:
- (a) \( A \cap B \): Shade the region where \( A \) and \( B \) overlap.
- (b) \( A \cap C \): Shade the region where \( A \) and \( C \) overlap.
- (c) \( (A \cap B) \cup (A \cap C) \): Shade the union of the regions where \( A \) overlaps with both \( B \) and \( C \).
- (d) \( B \cup C \): Shade the entire region covered by \( B \) and \( C \), including their overlap.
- (e) \( A \cap (B \cup C) \): Shade the region where \( A \) overlaps with the union of \( B \) and \( C \).

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Problem 2: Set Operations

#### Task:
Given the universal set \( U = \{1, 2, 3, \ldots, 18, 19, 20\} \) and subsets:
\[
A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, \quad B = \{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\},
\]
\[
C = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}, \quad D = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\},
\]
list all elements of the following sets.

#### Solution:
1. (a) \( A \cap B \):
\[
A \cap B = \{5, 6, 7, 8, 9, 10\}
\]

2. (b) \( \overline{D} \cap A \):
\[
\overline{D} = U \setminus D = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}
\]
\[
\overline{D} \cap A = \{2, 4, 6, 8, 10\}
\]

3. (c) \( A \cap C \):
\[
A \cap C = \{2, 4, 6, 8, 10\}
\]

4. (d) \( \overline{D} \cap A \):
This is the same as part (b):
\[
\overline{D} \cap A = \{2, 4, 6, 8, 10\}
\]

5. (e) \( (A \cap B) \cup (A \cap C) \):
\[
A \cap B = \{5, 6, 7, 8, 9, 10\}, \quad A \cap C = \{2, 4, 6, 8, 10\}
\]
\[
(A \cap B) \cup (A \cap C) = \{2, 4, 5, 6, 7, 8, 9, 10\}
\]

6. (f) \( B + C \):
Assuming \( B + C \) means \( B \cup C \):
\[
B \cup C = \{2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20\}
\]

7. (g) \( B \cdot C \):
Assuming \( B \cdot C \) means \( B \cap C \):
\[
B \cap C = \{6, 8, 10, 12, 14\}
\]

8. (h) \( A \cdot (B + C) \):
Assuming \( B + C \) means \( B \cup C \):
\[
B \cup C = \{2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20\}
\]
\[
A \cdot (B \cup C) = A \cap (B \cup C) = \{2, 4, 5, 6, 7, 8, 9, 10\}
\]

9. (i) \( \overline{A} \cdot B \):
\[
\overline{A} = U \setminus A = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}
\]
\[
\overline{A} \cdot B = \overline{A} \cap B = \{11, 12, 13, 14, 15\}
\]

10. (j) \( \overline{A} \cdot B \):
This is the same as part (i):
\[
\overline{A} \cdot B = \{11, 12, 13, 14, 15\}
\]

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Problem 3: Venn Diagram and Survey Data

#### Task:
Use a Venn diagram to represent survey data about what people ate at a Superbowl party. Then determine how many had nothing.

#### Solution:
1. Define Sets:
- Let \( N \) be the set of people who had nachos.
- Let \( W \) be the set of people who had wings.
- Let \( P \) be the set of people who had potato skins.

2. Given Data:
- \( |N| = 48 \)
- \( |W| = 39 \)
- \( |P| = 35 \)
- \( |W \cap P| = 20 \)
- \( |P \cap N| = 19 \)
- \( |W \cap N| = 22 \)
- \( |W \cap N \cap P| = 10 \)

3. Venn Diagram:
- Start by placing \( |W \cap N \cap P| = 10 \) in the center.
- Calculate the remaining intersections:
\[
|W \cap N|_{\text{only}} = 22 - 10 = 12
\]
\[
|P \cap N|_{\text{only}} = 19 - 10 = 9
\]
\[
|W \cap P|_{\text{only}} = 20 - 10 = 10
\]
- Calculate the individual sets:
\[
|N|_{\text{only}} = 48 - 10 - 12 - 9 = 17
\]
\[
|W|_{\text{only}} = 39 - 10 - 12 - 10 = 7
\]
\[
|P|_{\text{only}} = 35 - 10 - 9 - 10 = 6
\]

4. Total Number of People:
\[
\text{Total} = 10 + 12 + 9 + 10 + 17 + 7 + 6 = 71
\]

5. People Who Had Nothing:
\[
\text{People who had nothing} = 90 - 71 = 19
\]

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Problem 4: True or False

#### Task:
Determine whether the given statements are true or false and explain why.

#### Solution:
1. (a) \( \{1, 2, 3\} \subseteq \{3, 2, 1, 4\} \):
- True. Every element of \( \{1, 2, 3\} \) is in \( \{3, 2, 1, 4\} \).

2. (b) \( \{3, 2, 1, 4\} \subseteq \{1, 2, 3\} \):
- False. The element \( 4 \) is in \( \{3, 2, 1, 4\} \) but not in \( \{1, 2, 3\} \).

3. (c) \( \emptyset \subseteq \{1, 2, 3\} \):
- True. The empty set is a subset of every set.

4. (d) \( 1 \in \{3, 2, 1, 4\} \):
- True. The element \( 1 \) is in the set \( \{3, 2, 1, 4\} \).

5. (e) \( \{1\} \in \{3, 2, 1, 4\} \):
- False. The set \( \{1\} \) is not an element of \( \{3, 2, 1, 4\} \); only individual elements are in the set.

6. (f) \( \{1\} \subseteq \{3, 2, 1, 4\} \):
- True. The set \( \{1\} \) is a subset of \( \{3, 2, 1, 4\} \).

7. (g) \( \{3, 2, 1, 4\} = \{1, 2, 3, 4\} \):
- True. Sets are equal if they contain the same elements, regardless of order.

8. (h) If \( A = \{1, 4, 3, 8, 2\} \), then \( |A| = 5 \):
- True. The set \( A \) has 5 distinct elements.

9. (i) If \( A = \{3, 2, 1, 4\} \), then \( \{8, 4\} \in 2^A \):
- False. \( \{8, 4\} \) is not a subset of \( A \) because \( 8 \notin A \).

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Final Answers:

1. Venn Diagrams: Shaded regions as described above.
2. Set Operations:
- (a) \( \{5, 6, 7, 8, 9, 10\} \)
- (b) \( \{2, 4, 6, 8, 10\} \)
- (c) \( \{2, 4, 6, 8, 10\} \)
- (d) \( \{2, 4, 6, 8, 10\} \)
- (e) \( \{2, 4, 5, 6, 7, 8, 9, 10\} \)
- (f) \( \{2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20\} \)
- (g) \( \{6, 8, 10, 12, 14\} \)
- (h) \( \{2, 4, 5, 6, 7, 8, 9, 10\} \)
- (i) \( \{11, 12, 13, 14, 15\} \)
- (j) \( \{11, 12, 13, 14, 15\} \)
3. Survey Data:
- Venn diagram as described.
- People who had nothing: \( \boxed{19} \)
4. True or False:
- (a) True
- (b) False
- (c) True
- (d) True
- (e) False
- (f) True
- (g) True
- (h) True
- (i) False

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Final Answer: \(\boxed{19}\)