Given sets $A = \{0, 2, 4, 6, 8, 10\}$, $B = \{1, 3, 5, 7, 9\}$, $C = \{1, 2, 4, 5, 7, 8\}$, and $D = \{1, 2, 3, 5, 7, 8, 9\}$, we are asked to find the following sets: (a) $(A - B) - C$ (b) $(A \cap B) \cup (A \cap C)$ (c) $D \cap (A \cap C)$ (e) $(A \cup B) - (C \cap D)$

1. Problem Description

Given sets $A = \{0, 2, 4, 6, 8, 10\}$, $B = \{1, 3, 5, 7, 9\}$, $C = \{1, 2, 4, 5, 7, 8\}$, and $D = \{1, 2, 3, 5, 7, 8, 9\}$, we are asked to find the following sets:

(a) $(A - B) - C$

(b) $(A \cap B) \cup (A \cap C)$

(c) $D \cap (A \cap C)$

(e) $(A \cup B) - (C \cap D)$

2. Solution Steps

(a) $(A - B) - C$

$A - B$ is the set of elements in $A$ that are not in $B$. Since $A$ contains only even numbers and $B$ contains only odd numbers, $A - B = A = \{0, 2, 4, 6, 8, 10\}$.

$(A - B) - C = A - C$ is the set of elements in $A$ that are not in $C$.

$A - C = \{0, 2, 4, 6, 8, 10\} - \{1, 2, 4, 5, 7, 8\} = \{0, 6, 10\}$

(b) $(A \cap B) \cup (A \cap C)$

$A \cap B$ is the set of elements that are in both $A$ and $B$. Since $A$ contains only even numbers and $B$ contains only odd numbers, $A \cap B = \emptyset$.

$A \cap C$ is the set of elements that are in both $A$ and $C$.

$A \cap C = \{0, 2, 4, 6, 8, 10\} \cap \{1, 2, 4, 5, 7, 8\} = \{2, 4, 8\}$.

$(A \cap B) \cup (A \cap C) = \emptyset \cup \{2, 4, 8\} = \{2, 4, 8\}$.

(c) $D \cap (A \cap C)$

We already found that $A \cap C = \{2, 4, 8\}$.

$D \cap (A \cap C) = D \cap \{2, 4, 8\} = \{1, 2, 3, 5, 7, 8, 9\} \cap \{2, 4, 8\} = \{2, 8\}$.

(e) $(A \cup B) - (C \cap D)$

$A \cup B$ is the set of elements that are in $A$ or $B$ or both.

$A \cup B = \{0, 2, 4, 6, 8, 10\} \cup \{1, 3, 5, 7, 9\} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.

$C \cap D$ is the set of elements that are in both $C$ and $D$.

$C \cap D = \{1, 2, 4, 5, 7, 8\} \cap \{1, 2, 3, 5, 7, 8, 9\} = \{1, 2, 5, 7, 8\}$.

$(A \cup B) - (C \cap D) = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} - \{1, 2, 5, 7, 8\} = \{0, 3, 4, 6, 9, 10\}$.

3. Final Answer

(a) $\{0, 6, 10\}$

(b) $\{2, 4, 8\}$

(c) $\{2, 8\}$

(e) $\{0, 3, 4, 6, 9, 10\}$