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.