Given the sets $A = \{1, 2, 3, 4, 5, 6\}$, $B = \{2, 4, 6\}$, $C = \{1, 2, 3\}$, $D = \{7, 8, 9\}$ and the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, we need to find the following: (1) $(B \cup C)^c$ (2) $A \setminus B$ (3) $(D \cap C^c) \cup (A \cap B)^c$

1. Problem Description

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

(1) $(B \cup C)^c$

(2) $A \setminus B$

(3) $(D \cap C^c) \cup (A \cap B)^c$

2. Solution Steps

(1) Finding $(B \cup C)^c$:

First, we find the union of $B$ and $C$:

$B \cup C = \{2, 4, 6\} \cup \{1, 2, 3\} = \{1, 2, 3, 4, 6\}$.

Next, we find the complement of $B \cup C$ with respect to the universal set $U$:

$(B \cup C)^c = U \setminus (B \cup C) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \setminus \{1, 2, 3, 4, 6\} = \{5, 7, 8, 9, 10\}$.

(2) Finding $A \setminus B$:

$A \setminus B$ represents the elements in $A$ that are not in $B$.

$A \setminus B = \{1, 2, 3, 4, 5, 6\} \setminus \{2, 4, 6\} = \{1, 3, 5\}$.

(3) Finding $(D \cap C^c) \cup (A \cap B)^c$:

First, find $C^c$:

$C^c = U \setminus C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \setminus \{1, 2, 3\} = \{4, 5, 6, 7, 8, 9, 10\}$.

Next, find $D \cap C^c$:

$D \cap C^c = \{7, 8, 9\} \cap \{4, 5, 6, 7, 8, 9, 10\} = \{7, 8, 9\}$.

Now, find $A \cap B$:

$A \cap B = \{1, 2, 3, 4, 5, 6\} \cap \{2, 4, 6\} = \{2, 4, 6\}$.

Next, find $(A \cap B)^c$:

$(A \cap B)^c = U \setminus (A \cap B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \setminus \{2, 4, 6\} = \{1, 3, 5, 7, 8, 9, 10\}$.

Finally, find $(D \cap C^c) \cup (A \cap B)^c$:

$(D \cap C^c) \cup (A \cap B)^c = \{7, 8, 9\} \cup \{1, 3, 5, 7, 8, 9, 10\} = \{1, 3, 5, 7, 8, 9, 10\}$.

3. Final Answer

(1) $(B \cup C)^c = \{5, 7, 8, 9, 10\}$

(2) $A \setminus B = \{1, 3, 5\}$

(3) $(D \cap C^c) \cup (A \cap B)^c = \{1, 3, 5, 7, 8, 9, 10\}$