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### **Exercise 2:**
**Use set-builder notation to describe the sets:**

**a) {0, 3, 6, 9, 12}**

This set contains multiples of 3, starting from 0 up to 12. In set-builder notation:
\[
\{ x \in \mathbb{N} \ | \ x = 3n, \ n \in \mathbb{N} \text{ and } 0 \leq x \leq 12 \}
\]
where \( \mathbb{N} \) is the set of natural numbers.

**b) \{-3, -2, -1, 0, 1, 2, 3\}**

This set contains integers between -3 and 3 (inclusive). In set-builder notation:
\[
\{ x \in \mathbb{Z} \ | \ -3 \leq x \leq 3 \}
\]
where \( \mathbb{Z} \) is the set of integers.

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### **Exercise 3:**
**Determine whether the pairs of sets are equal:**

**a) \{1, 3, 3, 3, 5, 5, 5, 5, 5\} and \{5, 3, 1\}**

Yes, these sets are equal because sets ignore duplicates and order. Both simplify to:
\[
\{1, 3, 5\}.
\]

**b) \{\{1\}\} and \{1, \{1\}\}**

No, these sets are not equal. The first set contains only one element, \(\{1\}\), while the second set contains two elements: \(1\) and \(\{1\}\).

**c) \(\emptyset\) and \(\{\emptyset\}\)**

No, these sets are not equal. The first set is the empty set with no elements, while the second set contains one element, which is the empty set.

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### **Exercise 4:**
**Determine subsets:**

Sets given:
\[
A = \{2, 4, 6\}, \ B = \{2, 6\}, \ C = \{4, 6\}, \ D = \{4, 6, 8\}.
\]

1. \( B \subseteq A \): True (all elements of \(B\) are in \(A\)).
2. \( C \subseteq A \): True (all elements of \(C\) are in \(A\)).
3. \( D \subseteq A \): False (\(8 \notin A\)).

Additionally:
- \( B \subseteq D \): False (\(2 \notin D\)).
- \( C \subseteq D \): True (all elements of \(C\) are in \(D\)).

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### **Exercise 5:**
**Determine if 2 or \(\{2\}\) is an element of the set:**

**a) \(\{x \in \mathbb{R} : x \text{ is an integer greater than 1}\}\)**  
- \(2 \in \text{set}\), because \(2\) is an integer greater than 1.  
- \(\{2\} \notin \text{set}\), because \(\{2\}\) is not an integer.

**b) \(\{x \in \mathbb{R} : x \text{ is the square of an integer}\}\)**  
- \(2 \notin \text{set}\), because \(2\) is not the square of any integer.  
- \(\{2\} \notin \text{set}\), because \(\{2\}\) is not a number.

**c) \(\{2, \{2\}\}\)**  
- \(2 \in \text{set}\), because \(2\) is explicitly listed.  
- \(\{2\} \in \text{set}\), because \(\{2\}\) is explicitly listed.

**d) \(\{\{2\}, \{\{2\}\}\}\)**  
- \(2 \notin \text{set}\), because \(2\) is not explicitly listed.  
- \(\{2\} \in \text{set}\), because \(\{2\}\) is explicitly listed.  

**e) \(\{2, \{2\}, \{2, \{2\}\}\}\)**  
- \(2 \in \text{set}\), because \(2\) is explicitly listed.  
- \(\{2\} \in \text{set}\), because \(\{2\}\) is explicitly listed.  

**f) \(\{\{\{2\}\}\}\)**  
- \(2 \notin \text{set}\), because \(2\) is not explicitly listed.  
- \(\{2\} \notin \text{set}\), because \(\{2\}\) is not explicitly listed.  

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Do you need further clarifications or additional explanations?  

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### Related Questions:
1. How do duplicates and order affect equality in sets?
2. Can you explain the difference between \(\emptyset\) and \(\{\emptyset\}\) in more detail?
3. Why is \(\{1\}\) different from \(1\) in a set?
4. What are the conditions for a set to be a subset of another?
5. How does set-builder notation vary for different types of numbers?

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### Tip:
Remember, sets do not care about order or repetition, but they do distinguish between elements and subsets!

Solve exercises related to sets, including set-builder notation, subset determination, and set equality from the given problems.

Explore set theory problems including set-builder notation, determining subsets, and set equality. Includes step-by-step explanations for Grades 9-12, covering key concepts like duplicates, subsets, and empty sets.

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