We are given four sets: $A = \{0, 2, 4, 6, 8, 10\}$ $B = \{1, 3, 5, 7, 9\}$ $C = \{1, 2, 4, 5, 7, 8\}$ $D = \{1, 2, 3, 5, 7, 8, 9\}$ We need 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

We are given four sets:

$A = \{0, 2, 4, 6, 8, 10\}$

$B = \{1, 3, 5, 7, 9\}$

$C = \{1, 2, 4, 5, 7, 8\}$

$D = \{1, 2, 3, 5, 7, 8, 9\}$

We need 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) $(A - B) - C$

First, we find $A - B$, which is the set of elements in $A$ but not in $B$.

$A - B = \{0, 2, 4, 6, 8, 10\}$

Then we find $(A - B) - C$, which is the set of elements in $A - B$ but not in $C$.

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

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

First, we find $A \cap B$, which is the set of elements in both $A$ and $B$.

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

Then, we find $A \cap C$, which is the set of elements in both $A$ and $C$.

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

Finally, we find $(A \cap B) \cup (A \cap C)$, which is the union of the two sets.

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

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

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

Now, we find $D \cap (A \cap C)$, which is the set of elements in both $D$ and $A \cap C$.

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

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

First, we find $A \cup B$, which is the set of elements in either $A$ or $B$ or both.

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

Then, we find $C \cap D$, which is the set of elements in both $C$ and $D$.

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

Finally, we find $(A \cup B) - (C \cap D)$, which is the set of elements in $A \cup B$ but not in $C \cap D$.

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

3. Final Answer

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

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

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

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