Let's solve the problems in this assignment step by step:

1. Verify the given set theory identity:

Given:

• $A = \{1, 3, 5\}$
• $B = \{2, 4, 6, 8\}$
• $C = \{2, 5, 10\}$
• $U = \{1, 2, 3, 4, 5, 6, 8, 9, 10\}$

Verify:

1. $(A \cap B)^C = A^C \cup B^C$
2. $(A \cup B)^C = A^C \cap B^C$

Solution:

For the universal set $U$, the complement of any set $X$ is defined as: $X^C = U - X$

(a) $(A \cap B)^C = A^C \cup B^C$

1. Compute $A \cap B = \emptyset$ (no common elements between $A$ and $B$).
   So, $(A \cap B)^C = U - \emptyset = U = \{1, 2, 3, 4, 5, 6, 8, 9, 10\}$.

2. Compute $A^C = U - A = \{2, 4, 6, 8, 9, 10\}$.
   Compute $B^C = U - B = \{1, 3, 5, 9, 10\}$.
   Compute $A^C \cup B^C = \{2, 4, 6, 8, 9, 10\} \cup \{1, 3, 5, 9, 10\} = U$.

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

1. Compute $A \cup B = \{1, 2, 3, 4, 5, 6, 8\}$.
   So, $(A \cup B)^C = U - (A \cup B) = \{9, 10\}$.

2. From above:

   • $A^C = \{2, 4, 6, 8, 9, 10\}$
   • $B^C = \{1, 3, 5, 9, 10\}$
     Compute $A^C \cap B^C = \{9, 10\}$.

Hence, both identities are verified.

2. Composition of Functions

Given:

• $f(x) = x + 1$
• $g(x) = 2x$

Find:

1. $f \circ g(x)$
2. $g \circ f(x)$

Solution:

1. $f \circ g(x) = f(g(x)) = f(2x) = 2x + 1$.
2. $g \circ f(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2$.

3. Adjacency Matrix of the Graph

The graph has 5 vertices ($V_1, V_2, V_3, V_4, V_5$) with edges as indicated.

Adjacency Matrix:

The adjacency matrix $A$ is a 5x5 matrix where $A[i][j] = 1$ if there is an edge between $V_i$ and $V_j$, otherwise $A[i][j] = 0$.

From the graph:

• $V_1$ is connected to $V_2, V_3$
• $V_2$ is connected to $V_1, V_3, V_5$
• $V_3$ is connected to $V_1, V_2, V_4$
• $V_4$ is connected to $V_3, V_5$
• $V_5$ is connected to $V_2, V_4$