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.