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Find the smallest element in each of these subset of N

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### Extracted Exercises:
1. \( A = \{n \in \mathbb{N} : n \text{ is even}\} \)
2. \( B = \{n \in \mathbb{N} : 3 \leq n \leq 10\} \)
3. \( C = \{n \in \mathbb{N} : 1 \leq n^2 \leq 25\} \)
4. \( D = \{n \in \mathbb{N} : n \text{ is a multiple of 5}\} \)
5. \( E = \{n \in \mathbb{N} : n \text{ is a prime number}\} \)

### Solutions:
1. **Set A**: The set of even natural numbers starts from 2, so the smallest element in set \( A \) is **2**.
   
2. **Set B**: This set includes all natural numbers from 3 to 10, so the smallest element in set \( B \) is **3**.

3. **Set C**: Here, \( n^2 \) must be between 1 and 25, so \( n \) can be 1, 2, 3, 4, or 5. The smallest element is **1** (since \( 1^2 = 1 \)).

4. **Set D**: This set contains multiples of 5, starting from 5, so the smallest element in set \( D \) is **5**.

5. **Set E**: The prime numbers start from 2, so the smallest element in set \( E \) is **2**.

Would you like more details on these solutions, or do you have any questions? Here are some related questions to deepen your understanding:

1. What are the properties of natural numbers, and how are they defined?
2. Can you find the next smallest element in each of these sets?
3. How do you define a prime number, and why is 2 considered the smallest prime?
4. What would be the smallest element if \( n \) was an odd natural number in set \( A \)?
5. What if the set \( D \) was restricted to multiples of 5 that are also prime numbers?

**Tip:** When dealing with subsets, always consider the defining properties of the set elements to determine which elements meet the conditions, especially when finding the smallest or largest element.

Discover how to find the smallest element in subsets of natural numbers such as even numbers, numbers between 3 and 10, perfect squares between 1 and 25, multiples of 5, and prime numbers. Get detailed solutions and deepen your understanding of set theory and number properties. Suitable for students in Grades 4-6.

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