iven three sets P, Q and R such that: P = {x: x is a natural number between 10 and 16},
Q = {y: y is a even number between 8 and 20} and R = {7, 9, 11, 14, 18, 20}
(i) Find the difference of two sets P and Q (ii) Find Q - R
(iii) Find R – P (iv) Find Q – P

Given the sets:
- \( P = \{x: x \text{ is a natural number between 10 and 16}\} \)
- \( Q = \{y: y \text{ is an even number between 8 and 20}\} \)
- \( R = \{7, 9, 11, 14, 18, 20\} \)

Let's first list the elements of each set:

- \( P = \{10, 11, 12, 13, 14, 15, 16\} \)
- \( Q = \{8, 10, 12, 14, 16, 18, 20\} \)
- \( R = \{7, 9, 11, 14, 18, 20\} \)

Now, let's solve each part:

### (i) \( P - Q \) (Difference of \( P \) and \( Q \))
The difference of two sets \( P \) and \( Q \), denoted by \( P - Q \), is the set of elements that are in \( P \) but not in \( Q \).

\[ P - Q = \{x: x \in P \text{ and } x \notin Q\} \]

Comparing the elements in \( P \) with those in \( Q \):
- \( P = \{10, 11, 12, 13, 14, 15, 16\} \)
- \( Q = \{8, 10, 12, 14, 16, 18, 20\} \)

The elements in \( P \) that are not in \( Q \) are \( \{11, 13, 15\} \).

So, \( P - Q = \{11, 13, 15\} \).

### (ii) \( Q - R \) (Difference of \( Q \) and \( R \))
The difference \( Q - R \) is the set of elements that are in \( Q \) but not in \( R \).

\[ Q - R = \{x: x \in Q \text{ and } x \notin R\} \]

Comparing the elements in \( Q \) with those in \( R \):
- \( Q = \{8, 10, 12, 14, 16, 18, 20\} \)
- \( R = \{7, 9, 11, 14, 18, 20\} \)

The elements in \( Q \) that are not in \( R \) are \( \{8, 10, 12, 16\} \).

So, \( Q - R = \{8, 10, 12, 16\} \).

### (iii) \( R - P \) (Difference of \( R \) and \( P \))
The difference \( R - P \) is the set of elements that are in \( R \) but not in \( P \).

\[ R - P = \{x: x \in R \text{ and } x \notin P\} \]

Comparing the elements in \( R \) with those in \( P \):
- \( R = \{7, 9, 11, 14, 18, 20\} \)
- \( P = \{10, 11, 12, 13, 14, 15, 16\} \)

The elements in \( R \) that are not in \( P \) are \( \{7, 9, 18, 20\} \).

So, \( R - P = \{7, 9, 18, 20\} \).

### (iv) \( Q - P \) (Difference of \( Q \) and \( P \))
The difference \( Q - P \) is the set of elements that are in \( Q \) but not in \( P \).

\[ Q - P = \{x: x \in Q \text{ and } x \notin P\} \]

Comparing the elements in \( Q \) with those in \( P \):
- \( Q = \{8, 10, 12, 14, 16, 18, 20\} \)
- \( P = \{10, 11, 12, 13, 14, 15, 16\} \)

The elements in \( Q \) that are not in \( P \) are \( \{8, 18, 20\} \).

So, \( Q - P = \{8, 18, 20\} \).

### Summary:
- (i) \( P - Q = \{11, 13, 15\} \)
- (ii) \( Q - R = \{8, 10, 12, 16\} \)
- (iii) \( R - P = \{7, 9, 18, 20\} \)
- (iv) \( Q - P = \{8, 18, 20\} \)

Would you like any further explanation on these steps?

Here are some related questions:
1. How do you find the union of two sets?
2. What is the intersection of sets \( P \) and \( Q \)?
3. How would you find \( P \cup R \) (the union of \( P \) and \( R \))?
4. What is the symmetric difference of \( P \) and \( Q \)?
5. How can you represent these set operations using Venn diagrams?

**Tip:** Remember that the difference between two sets \( A \) and \( B \) (denoted \( A - B \)) is not the same as \( B - A \); it's essential to keep track of the order in which sets are subtracted.

Explore set theory with the difference of sets P, Q, and R. Understand how to find P - Q, Q - R, R - P, and Q - P using detailed step-by-step solutions. Suitable for advanced high school students.

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