The document defines sets U, B, A, C and their elements. It then performs various set operations on these sets including intersection, union, complement and displays the results. It calculates (A ∩ B), (A ∩ C), (B ∩ C), their complements and unions.

1. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
𝑼 𝑨
𝑩 𝑪

2. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
𝑨ˊ
𝑩ˊ 𝑪ˊ
𝑪ˊ = 𝟏, 𝟐, 𝟔, 𝟖, 𝟗, 𝟏𝟎
𝑩ˊ = 𝟏, 𝟓, 𝟔, 𝟕, 𝟗, 𝟏𝟎
𝑨ˊ = 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎

3. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
(𝑨 ∩ 𝑩) ∩ 𝑪 = 𝟑
(𝑨 ∩ 𝑩) ∩ 𝑪ˊ = 𝟐, 𝟑 ∩ 𝟏, 𝟐, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟐
(𝑨 ∩ 𝑪) ∩ 𝑩ˊ = 𝟑, 𝟕 ∩ 𝟏, 𝟓, 𝟔, 𝟕, 𝟗, 𝟏𝟎 = 𝟕
(𝑩 ∩ 𝑪) ∩ 𝑨ˊ = 𝟑, 𝟒 ∩ 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟒
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∩ 𝑩) ∩ 𝑪 (𝑨 ∩ 𝑩) ∩ 𝑪ˊ
(𝑨 ∩ 𝑪) ∩ 𝑩ˊ (𝑩 ∩ 𝑪) ∩ 𝑨ˊ

4. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
𝑨 ∪ 𝑩 ∩ 𝑪 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟕, 𝟖 ∩ 𝟑, 𝟒, 𝟓, 𝟕 = 𝟑, 𝟒, 𝟕
(𝑨 ∪ 𝑩) ∩ 𝑪ˊ = 𝟏, 𝟐, 𝟑, 𝟒, 𝟕, 𝟖 ∩ 𝟏, 𝟐, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟏, 𝟐, 𝟖
(𝑨 ∪ 𝑪) ∩ 𝑩ˊ = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟕 ∩ 𝟏, 𝟓, 𝟔, 𝟕, 𝟗, 𝟏𝟎 = 𝟏, 𝟓, 𝟕
(𝑩 ∪ 𝑪) ∩ 𝑨ˊ = 𝟐, 𝟑, 𝟒, 𝟓, 𝟕, 𝟖 ∩ 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟒, 𝟓, 𝟖
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∪ 𝑩) ∩ 𝑪 (𝑨 ∪ 𝑩) ∩ 𝑪ˊ
(𝑨 ∪ 𝑪) ∩ 𝑩ˊ (𝑩 ∪ 𝑪) ∩ 𝑨ˊ

5. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
A ∩ B= 𝟐, 𝟑
𝑨 ∩ 𝑪 = 𝟑, 𝟕
B∩𝐂 = 𝟑, 𝟒
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∩ 𝑩)
(𝑨 ∩ 𝑪) (𝑩 ∩ 𝑪)

6. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
(A ∩ B)՛= 𝟏, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
(𝑨 ∩ 𝑪)՛ = 𝟏, 𝟐, 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎
(B∩𝐂) = 𝟏, 𝟐, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∩ 𝑩)՛
(𝑨 ∩ 𝑪)՛ (𝑩 ∩ 𝑪)՛

7. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
A ∪ B= 𝟏, 𝟐, 𝟑, 𝟒, 𝟕, 𝟖
𝑨 ∪ 𝑪 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟕
B ∪ 𝐂 = 𝟐, 𝟑, 𝟒, 𝟓, 𝟕, 𝟖
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∪ 𝑩)
(𝑨 ∪ 𝑪) (𝑩 ∪ 𝑪)

8. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
(A ∪ B)՛= 𝟓, 𝟔, 𝟗, 𝟏𝟎
(𝑨 ∪ 𝑪)՛ = 𝟔, 𝟖, 𝟗, 𝟏𝟎
(B ∪ 𝐂)՛ = 𝟏, 𝟔, 𝟗, 𝟏𝟎
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∪ 𝑩)՛
(𝑨 ∪ 𝑪)՛ (𝑩 ∪ 𝑪)՛

9. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
A՛∪ B՛= 𝑨 ∩ 𝑩 ՛ = 𝟏, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑨՛ ∪ 𝑪՛ = 𝑨 ∩ 𝑪 ՛ = 𝟏, 𝟐, 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎
B՛ ∪ 𝐂՛ = 𝑩 ∩ 𝑪 ՛ = 𝟏, 𝟐, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A՛∪ B՛ = 𝑨 ∩ 𝑩 ՛
𝑨՛ ∪ 𝑪՛ = 𝑨 ∩ 𝑪 ՛ B՛ ∪ 𝐂՛ = 𝑩 ∩ 𝑪 ՛
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10

10. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
A՛∩ B՛= 𝑨 ∪ 𝑩 ՛ = 𝟓, 𝟔, 𝟗, 𝟏𝟎
𝑨՛ ∩ 𝑪՛ = 𝑨 ∪ 𝑪 ՛ = 𝟔, 𝟖, 𝟗, 𝟏𝟎
B՛ ∩ 𝐂՛ = 𝑩 ∪ 𝑪 ՛ = 𝟏, 𝟔, 𝟗, 𝟏𝟎
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A՛∩ B՛ = 𝑨 ∪ 𝑩 ՛
𝑨՛ ∩ 𝑪՛ = 𝑨 ∪ 𝑪 ՛ B՛ ∩ 𝐂՛ = 𝑩 ∪ 𝑪 ՛
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10

11. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
𝑨 ∪ 𝑩 ∩ 𝑪 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟕, 𝟖 ∩ 𝟑, 𝟒, 𝟓, 𝟕 = 𝟑, 𝟒, 𝟕
(𝑨 ∪ 𝑪) ∩ 𝑩 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟕 ∩ 𝟐, 𝟑, 𝟒, 𝟖 = 𝟐, 𝟑, 𝟒,
(𝑩 ∪ 𝑪) ∩ 𝑨 = 𝟐, 𝟑, 𝟒, 𝟓, 𝟕, 𝟖 ∩ 𝟏, 𝟐, 𝟑, 𝟕 = 𝟐, 𝟑, 𝟕
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑨 ∪ 𝑩) ∩ 𝑪 (𝑨 ∪ 𝑪) ∩ 𝑩
(𝑩 ∪ 𝑪) ∩ 𝑨

12. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
(𝑩 ∩ 𝑪) ∩ 𝑨՛ = 𝟑, 𝟒 ∩ 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟒
(𝑨 ∩ 𝑪) ∩ 𝑩՛ = 𝟑, 𝟕 ∩ 𝟏, 𝟓, 𝟔, 𝟕, 𝟗, 𝟏𝟎 = 𝟕
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
𝑨 ∩ 𝑩 ∩ 𝑪՛ (𝑩 ∩ 𝑪) ∩ 𝑨՛
(𝑨 ∩ 𝑪) ∩ 𝑩՛
𝑨 ∩ 𝑩 ∪ 𝑪՛ = 𝟐, 𝟑 ∪ 𝟏, 𝟐, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟏, 𝟐, 𝟑, 𝟔, 𝟖, 𝟗, 𝟏𝟎
A B
C
U
1 8
2
3
7 4
5
6,9,10
𝑨 ∩ 𝑩 ∪ 𝑪՛
𝑨 ∩ 𝑩 ∩ 𝑪՛ = 𝟐, 𝟑 ∩ 𝟏, 𝟐, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟐

13. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10
(A∩B)∪(A∩B)∪(A∩B) ՛
(A∩B)∪(A∩B)∪(A∩B) (𝑩 ∪ 𝑪)՛
A B
C
U
1 8
2
3
7 4
5
6,9,10
A∩(B∪C)՛= 𝟏, 𝟐, 𝟑, 𝟕 ∩ 𝟏, 𝟔, 𝟗, 𝟏𝟎 = 𝟏
A∩(B∪C)՛∪ 𝐁∩(A∪C)՛∪ C∩(A∪B)՛
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
B∩(A∪C)՛= 𝟐, 𝟑, 𝟒, 𝟖 ∩ 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟖
C∩(A∪B)՛= 𝟑, 𝟒, 𝟓, 𝟕 ∩ 𝟓, 𝟔, 𝟗, 𝟏𝟎 = 𝟓
(A∩B)∪(A∩B)∪(A∩B) ՛= 𝟏, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎
(A∩B)∪(A∩B)∪(A∩B) = 𝟐, 𝟑, 𝟒, 𝟕

14. 𝑼 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎
𝑩 = 𝟐, 𝟑, 𝟒, 𝟖
𝑨 = 𝟏, 𝟐, 𝟑, 𝟕
𝑪 = 𝟑, 𝟒, 𝟓, 𝟕
V
V
A B
C
U
1 2
3
7
8
4
5
6,9,10 (𝑨 ∩ 𝑩) ∩ 𝑪՛ ∪ 𝑩 ∩ 𝑪 ∩ 𝑨՛ ∪ 𝑨 ∩ 𝑪 ∩ 𝑩՛
(𝑩 ∩ 𝑪) ∩ 𝑨՛ ∪ A∩(B∪C)՛
A B
C
U
1 8
2
3
7 4
5
6,9,10
A B
C
U
1 8
2
3
7 4
5
6,9,10
(𝑩 ∩ 𝑪) ∩ 𝑨՛ = 𝟑, 𝟒 ∩ 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟒
(𝑨 ∩ 𝑪) ∩ 𝑩՛ = 𝟑, 𝟕 ∩ 𝟏, 𝟓, 𝟔, 𝟕, 𝟗, 𝟏𝟎 = 𝟕
𝑨 ∩ 𝑩 ∩ 𝑪՛ = 𝟐, 𝟑 ∩ 𝟏, 𝟐, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟐
(𝑨 ∩ 𝑩) ∩ 𝑪՛ ∪ 𝑩 ∩ 𝑪 ∩ 𝑨՛ ∪ 𝑨 ∩ 𝑪 ∩ 𝑩՛ = 𝟐, 𝟒, 𝟕
(𝑩 ∩ 𝑪) ∩ 𝑨՛ = 𝟑, 𝟒 ∩ 𝟒, 𝟓, 𝟔, 𝟖, 𝟗, 𝟏𝟎 = 𝟒
A∩(B∪C)՛= 𝟏, 𝟐, 𝟑, 𝟕 ∩ 𝟏, 𝟔, 𝟗, 𝟏𝟎 = 𝟏
(𝑩 ∩ 𝑪) ∩ 𝑨՛ ∪ A∩(B∪C)՛ = 𝟏, 𝟒

15. More than 250 combinations of shadings are possible for three circle
problems as given by the sum of combinations:
𝑛𝐶𝑟 =
𝑛!
𝑛 − 𝑟 ! 𝑟!
n r nCr
8 1 8
8 2 28
8 3 56
8 4 70
8 5 56
8 6 28
8 7 8
8 8 1
255