https://file.aiquickdraw.com/mathbot/file/17260843736693afs7lzx.jpg

Let's break down the question from the image and analyze the two sets provided:

- **Set A:** \( A = \{ x \mid x \in \mathbb{N} \text{ and } 99 \leq x \leq 104 \} \)
- **Set B:** \( B = \{ x \mid x \in \mathbb{N} \text{ and } 98 < x < 105 \} \)

### 1. Defining the Sets:
- Set \( A \) includes natural numbers from 99 to 104, inclusive. Therefore, \( A = \{99, 100, 101, 102, 103, 104\} \).
- Set \( B \) includes natural numbers between 98 and 105, excluding 98 and 105. Therefore, \( B = \{99, 100, 101, 102, 103, 104\} \).

### 2. Checking for Equivalence:
Two sets are said to be **equivalent** if they have the same number of elements. In this case:
- The number of elements in \( A \) is 6.
- The number of elements in \( B \) is also 6.

Since \( n(A) = n(B) = 6 \), the sets are **equivalent**.

### 3. Checking for Equality:
Two sets are **equal** if they contain exactly the same elements. Here, both \( A \) and \( B \) contain the same elements: \( \{99, 100, 101, 102, 103, 104\} \).

Thus, \( A = B \), meaning the sets are **equal**.

### Answer to Part a:
The sets are **equivalent** because \( n(A) = n(B) \). So, the correct option is:
- **Option A**: The sets are equivalent because \( n(A) = n(B) \).

### Answer to Part b:
The sets are **equal** because they contain the exact same elements. Therefore, the answer would be:
- Yes, the sets are equal.

Would you like further details on set theory or have any additional questions? Here are some related questions you might find interesting:

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1. What is the difference between equivalent and equal sets?
2. How do you determine the cardinality of a set?
3. Can two sets with different numbers of elements ever be equivalent?
4. What are the properties of natural numbers used in set theory?
5. How does the inclusion of inequality symbols impact set notation?

**Tip:** Always check both the elements and the cardinality of a set when determining equality and equivalence!

Are the sets equivalent? Are the sets equal? A = {x | x ∈ N and 99 ≤ x ≤ 104}, B = {x | x ∈ N and 98 < x < 105}

Understand the difference between equivalent and equal sets with this problem, featuring two sets A = {x | 99 ≤ x ≤ 104} and B = {x | 98 < x < 105}. This lesson explains how to determine whether two sets are equivalent based on cardinality and whether they are equal based on their elements.

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