Given that E= (x:1<x<30, x is a positive integer), P = ( x:x is an odd number) Q= x:15 <x<27 and x is a multiple of 3) and R = (x:x is a factor of 45 

We are given the following sets:

- **Set E**: \( E = \{x : 1 < x < 30, x \text{ is a positive integer}\} \)  
  This means that set \( E \) includes all integers between 2 and 29.
  
  So, \( E = \{2, 3, 4, 5, 6, \dots, 29\} \).
  
- **Set P**: \( P = \{x : x \text{ is an odd number}\} \)  
  This set consists of all odd numbers. Since the universal set here is \( E \), we focus on the odd numbers in \( E \).

  So, \( P \cap E = \{3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29\} \).

- **Set Q**: \( Q = \{x : 15 < x < 27, x \text{ is a multiple of 3}\} \)  
  The elements of \( Q \) are multiples of 3 between 16 and 26.

  So, \( Q = \{18, 21, 24\} \).

- **Set R**: \( R = \{x : x \text{ is a factor of 45}\} \)  
  The factors of 45 are the numbers that divide 45 evenly.

  So, \( R = \{1, 3, 5, 9, 15, 45\} \). Since the universal set is \( E \) (i.e., integers between 2 and 29), we take only those factors of 45 within this range.

  Hence, \( R \cap E = \{3, 5, 9, 15\} \).

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### Summary of the sets:
- \( E = \{2, 3, 4, 5, 6, \dots, 29\} \)
- \( P = \{3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29\} \)
- \( Q = \{18, 21, 24\} \)
- \( R = \{3, 5, 9, 15\} \)

Would you like me to perform any specific operations (like intersections, unions, etc.) on these sets, or provide a Venn diagram?

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### Related Questions:
1. What is the intersection of sets \( P \) and \( Q \)?
2. What is the union of sets \( E \), \( P \), and \( R \)?
3. Can you find the difference between sets \( P \) and \( R \)?
4. What is the cardinality of set \( Q \)?
5. How would you represent the sets in a Venn diagram?

### Tip:
When working with sets, always consider the universal set to keep the context clear, especially when performing operations like intersections or unions.

Given that E= (x:1<x<30, x is a positive integer), P = ( x:x is an odd number) Q= x:15 <x<27 and x is a multiple of 3) and R = (x:x is a factor of 45)

Explore a set theory problem involving positive integers between 2 and 29. This problem covers odd numbers, multiples of 3 between 15 and 27, and factors of 45. Learn how to find the intersection, union, and cardinality of these sets. Suitable for students in Grades 9-11.

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