We are given four sets: $A = \{1, 2, 3, 4, 5, 6\}$, $B = \{2, 4, 6\}$, $C = \{1, 2, 3\}$, and $D = \{7, 8, 9\}$. The universe set is $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. We need to find the complement of the union of sets $B$ and $C$, which is $(B \cup C)^c$.

1. Problem Description

We are given four sets: $A = \{1, 2, 3, 4, 5, 6\}$, $B = \{2, 4, 6\}$, $C = \{1, 2, 3\}$, and $D = \{7, 8, 9\}$. The universe set is $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. We need to find the complement of the union of sets $B$ and $C$, which is $(B \cup C)^c$.

2. Solution Steps

First, we find the union of sets $B$ and $C$.

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

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

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

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

$(B \cup C)^c$ is the set of all elements in $U$ that are not in $B \cup C$.

$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $B \cup C = \{1, 2, 3, 4, 6\}$.

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

3. Final Answer

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