Practice for students Grade 9 - SETS Theory Interactive Worksheet ... - Free Printable

Problem Analysis and Solution

The task involves solving problems related to set operations: intersection, union, and complements. Let's go through each problem step by step.

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#### Section 1: Basic Set Operations

##### Problem 1
- Given:
\( A = \{1, 3, 5, 9, 11\} \)
\( B = \{0, 1, 2, 3, 8, 9\} \)

- Tasks:
1. \( A \cap B \)
2. \( A \cup B \)

- Solution:
1. Intersection (\( A \cap B \)): Elements common to both \( A \) and \( B \).
\( A \cap B = \{1, 3, 9\} \)

2. Union (\( A \cup B \)): All elements in either \( A \) or \( B \).
\( A \cup B = \{0, 1, 2, 3, 5, 8, 9, 11\} \)

##### Problem 2
- Given:
\( A = \{2, 4, 6\} \)
\( B = \{1, 3, 5\} \)

- Task:
\( A \cap B \)

- Solution:
Intersection (\( A \cap B \)): No elements are common to both \( A \) and \( B \).
\( A \cap B = \emptyset \) (or \(\{\}\))

##### Problem 3
- Given:
\( A = \{-5, -2, 0, 2, 5, 9\} \)
\( B = \{\text{set of all whole numbers}\} \)

- Task:
\( A \cap B \)

- Solution:
Intersection (\( A \cap B \)): Whole numbers in \( A \) are \( \{0, 2, 5, 9\} \).
\( A \cap B = \{0, 2, 5, 9\} \)

##### Problem 4
- Given:
\( A = \{2, 4, 6\} \)
\( B = \{1, 3, 5\} \)

- Task:
\( A \cup B \)

- Solution:
Union (\( A \cup B \)): Combine all unique elements from \( A \) and \( B \).
\( A \cup B = \{1, 2, 3, 4, 5, 6\} \)

##### Problem 5
- Given:
\( A = \{-5, -2, 0, 2, 5, 9\} \)
\( B = \{\text{set of all whole numbers}\} \)

- Task:
\( A \cap B \)

- Solution:
Intersection (\( A \cap B \)): Whole numbers in \( A \) are \( \{0, 2, 5, 9\} \).
\( A \cap B = \{0, 2, 5, 9\} \)

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#### Section 2: Venn Diagram Problems

##### Problem 6
- Given:
Venn diagram with sets \( A \) and \( B \). Elements are labeled as follows:
- \( A \): \( \{2, 3, 4, 8\} \)
- \( B \): \( \{5, 7, 9\} \)
- Intersection (\( A \cap B \)): \( \{1\} \)
- Outside both sets: \( \{11\} \)

- Task:
\( A \cap B \)

- Solution:
Intersection (\( A \cap B \)): Elements in both \( A \) and \( B \). From the diagram, \( A \cap B = \{1\} \).

##### Problem 7
- Task:
\( A \cup B \)

- Solution:
Union (\( A \cup B \)): Combine all elements in \( A \) and \( B \), excluding duplicates.
\( A \cup B = \{1, 2, 3, 4, 5, 7, 8, 9\} \)

##### Problem 8
- Task:
\( A^C \) (Complement of \( A \))

- Solution:
Complement of \( A \): Elements outside \( A \) but within the universal set. From the diagram, the universal set includes \( \{1, 2, 3, 4, 5, 7, 8, 9, 11\} \). Elements not in \( A \) are \( \{5, 7, 9, 11\} \).
\( A^C = \{5, 7, 9, 11\} \)

##### Problem 9
- Task:
\( B^C \) (Complement of \( B \))

- Solution:
Complement of \( B \): Elements outside \( B \) but within the universal set. From the diagram, the universal set includes \( \{1, 2, 3, 4, 5, 7, 8, 9, 11\} \). Elements not in \( B \) are \( \{2, 3, 4, 8, 11\} \).
\( B^C = \{2, 3, 4, 8, 11\} \)

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#### Section 3: Universal Set and Complements

##### Problem 10
- Task:
Given the universal set is the integers from 1 through 100, describe the complement of \( \{2, 4, 6, 8, \ldots, 100\} \).

- Solution:
The given set is the set of all even integers from 1 to 100. The complement of this set within the universal set (integers from 1 to 100) is the set of all odd integers from 1 to 100.
\( \{1, 3, 5, 7, \ldots, 99\} \)

##### Problem 11
- Task:
Given the universal set is the integers from 1 through 27, describe the complement of \( \{1, 3, 5, 7, \ldots, 27\} \).

- Solution:
The given set is the set of all odd integers from 1 to 27. The complement of this set within the universal set (integers from 1 to 27) is the set of all even integers from 1 to 27.
\( \{2, 4, 6, 8, \ldots, 26\} \)

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#### Section 4: Advanced Set Operations

##### Problem 12
- Given:
\( A = \{-5, -4, 0, 4, 5\} \)
\( B = \{-6, -5, 0, 5, 6\} \)

- Task:
Intersection of \( A \) and \( B \)

- Solution:
Intersection (\( A \cap B \)): Elements common to both \( A \) and \( B \).
\( A \cap B = \{-5, 0, 5\} \)

##### Problem 13
- Task:
Union of \( A \) and \( B \)

- Solution:
Union (\( A \cup B \)): Combine all unique elements from \( A \) and \( B \).
\( A \cup B = \{-6, -5, -4, 0, 4, 5, 6\} \)

##### Problem 14
- Given:
\( A = \{-500, -12, -1, 1, 5, 57\} \)
\( B = \{-12, -1, 1, 5, 22\} \)

- Task:
Intersection of \( A \) and \( B \)

- Solution:
Intersection (\( A \cap B \)): Elements common to both \( A \) and \( B \).
\( A \cap B = \{-12, -1, 1, 5\} \)

##### Problem 15
- Task:
Union of \( A \) and \( B \)

- Solution:
Union (\( A \cup B \)): Combine all unique elements from \( A \) and \( B \).
\( A \cup B = \{-500, -12, -1, 1, 5, 22, 57\} \)

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Final Answers

1. \( A \cap B = \{1, 3, 9\} \)
2. \( A \cup B = \{0, 1, 2, 3, 5, 8, 9, 11\} \)
3. \( A \cap B = \emptyset \)
4. \( A \cup B = \{1, 2, 3, 4, 5, 6\} \)
5. \( A \cap B = \{0, 2, 5, 9\} \)
6. \( A \cap B = \{1\} \)
7. \( A \cup B = \{1, 2, 3, 4, 5, 7, 8, 9\} \)
8. \( A^C = \{5, 7, 9, 11\} \)
9. \( B^C = \{2, 3, 4, 8, 11\} \)
10. Complement of \( \{2, 4, 6, 8, \ldots, 100\} \): \( \{1, 3, 5, 7, \ldots, 99\} \)
11. Complement of \( \{1, 3, 5, 7, \ldots, 27\} \): \( \{2, 4, 6, 8, \ldots, 26\} \)
12. \( A \cap B = \{-5, 0, 5\} \)
13. \( A \cup B = \{-6, -5, -4, 0, 4, 5, 6\} \)
14. \( A \cap B = \{-12, -1, 1, 5\} \)
15. \( A \cup B = \{-500, -12, -1, 1, 5, 22, 57\} \)

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
&1. \{1, 3, 9\} \\
&2. \{0, 1, 2, 3, 5, 8, 9, 11\} \\
&3. \emptyset \\
&4. \{1, 2, 3, 4, 5, 6\} \\
&5. \{0, 2, 5, 9\} \\
&6. \{1\} \\
&7. \{1, 2, 3, 4, 5, 7, 8, 9\} \\
&8. \{5, 7, 9, 11\} \\
&9. \{2, 3, 4, 8, 11\} \\
&10. \{1, 3, 5, 7, \ldots, 99\} \\
&11. \{2, 4, 6, 8, \ldots, 26\} \\
&12. \{-5, 0, 5\} \\
&13. \{-6, -5, -4, 0, 4, 5, 6\} \\
&14. \{-12, -1, 1, 5\} \\
&15. \{-500, -12, -1, 1, 5, 22, 57\}
\end{aligned}
}
\]