Universal set U = {Positive real numbers less than 300}
A = {x ∈ N| x is divisible by 15}
B = {2, 4, 6, 8 ... 198, 200}
C = The set of all natural numbers divisible by 5
D = {x ∈ R| x < 100}
E = {3, 6, 9, 12...}

Which of the sets described above is an infinite set?

To determine which of the given sets is infinite, let's analyze each set:

1. **Set \( U \)**: 
   - \( U = \{ \text{Positive real numbers less than 300} \} \).
   - This set contains all positive real numbers less than 300. However, since it is limited by the number 300, \( U \) is **not infinite**.

2. **Set \( A \)**:
   - \( A = \{ x \in \mathbb{N} \mid x \text{ is divisible by 15} \} \).
   - This is the set of natural numbers divisible by 15. Given that \( U \) limits the elements to numbers less than 300, \( A \) would contain numbers like 15, 30, 45, ..., 285, which is a finite sequence. Thus, \( A \) is **not infinite**.

3. **Set \( B \)**:
   - \( B = \{ 2, 4, 6, 8, \dots, 198, 200 \} \).
   - This set is clearly a finite arithmetic sequence, with a last term of 200, so \( B \) is **not infinite**.

4. **Set \( C \)**:
   - \( C = \) The set of all natural numbers divisible by 5.
   - This set includes numbers like 5, 10, 15, 20, and so on, without any upper limit. Therefore, \( C \) is **infinite**.

5. **Set \( D \)**:
   - \( D = \{ x \in \mathbb{R} \mid x < 100 \} \).
   - This set includes all real numbers less than 100. However, it does not extend to infinity because it's limited to numbers less than 100, so \( D \) is **not infinite**.

6. **Set \( E \)**:
   - \( E = \{ 3, 6, 9, 12, \dots \} \).
   - This set is the sequence of multiples of 3, and since it doesn't have an upper bound, it is **infinite**.

**Conclusion**: The infinite sets among the described ones are **Set \( C \) and Set \( E \)**.

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Would you like any further details or explanations?

Here are some related questions you might find useful:

1. How do you determine if a set is infinite?
2. What is the difference between countably infinite and uncountably infinite sets?
3. Can a set of real numbers be finite?
4. How does the concept of a universal set limit the elements of a subset?
5. What are the properties of arithmetic sequences?
6. How do you find the number of elements in a finite set?
7. What are natural numbers and how do they differ from real numbers?
8. How can we describe the intersection of infinite sets?

**Tip:** Always check the definition and limits of a set to determine if it is finite or infinite.

Explore the concept of infinite sets in mathematics by identifying and analyzing specific sets such as natural numbers divisible by 5 and multiples of 3. Understand the definition and characteristics of infinite sets versus finite sets with clear examples and explanations.

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