Let's solve the questions step by step:

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1. Verify the Set Theory Properties

We need to verify:

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

Given Sets:

• $A = \{1, 3, 5\}$,
• $B = \{2, 4, 6, 8\}$,
• $C = \{2, 5, 10\}$,
• $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ (Universal Set).

Step 1: Verification of $(A \cap B)^C = A^C \cup B^C$:

1. Compute $A \cap B$:
   $A \cap B = \{\}$ (no common elements).

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

3. Compute $A^C$:
   $A^C = U - A = \{2, 4, 6, 7, 8, 9, 10\}$.

4. Compute $B^C$:
   $B^C = U - B = \{1, 3, 5, 7, 9, 10\}$.

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

Since $(A \cap B)^C = U$ and $A^C \cup B^C = U$, the property holds.

Step 2: Verification of $(A \cup B)^C = A^C \cap B^C$:

1. Compute $A \cup B$:
   $A \cup B = \{1, 2, 3, 4, 5, 6, 8\}$.

2. Compute $(A \cup B)^C$:
   $(A \cup B)^C = U - (A \cup B) = \{7, 9, 10\}$.

3. Compute $A^C \cap B^C$:
   $A^C \cap B^C = \{2, 4, 6, 7, 8, 9, 10\} \cap \{1, 3, 5, 7, 9, 10\} = \{7, 9, 10\}$.

Since $(A \cup B)^C = \{7, 9, 10\}$ and $A^C \cap B^C = \{7, 9, 10\}$, the property holds.

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2. Functions $f(x) = x + 1$, $g(y) = 2y$

We are tasked to find $f \circ g(x)$ and $g \circ f(x)$.

Step 1: Compute $f \circ g(x)$:

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

Step 2: Compute $g \circ f(x)$:

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

So:

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

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3. Adjacency Matrix

The problem involves finding the adjacency matrix for two graphs:

First Graph:

Vertices: $V_1, V_2, V_3, V_4, V_5$

Edges (connections):

• $V_1 \leftrightarrow V_2$,
• $V_1 \leftrightarrow V_3$,
• $V_1 \leftrightarrow V_4$,
• $V_3 \leftrightarrow V_4$,
• $V_5$ is isolated.

Adjacency matrix:

0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$ #### **Second Graph:** Vertices: $$V_1, V_2, V_3, V_4$$ Edges (connections): - $$V_1 \leftrightarrow V_2$$, - $$V_1 \leftrightarrow V_4$$, - $$V_2 \leftrightarrow V_3$$, - $$V_3 \leftrightarrow V_4$$. Adjacency matrix: $$\begin{bmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix}$$ --- Would you like me to explain any specific part in more detail? --- ### **Relative Questions:** 1. How does De Morgan’s Laws apply to complement operations in sets? 2. What are the differences between $$f \circ g(x)$$ and $$g \circ f(x)$$ in function composition? 3. How do you interpret an adjacency matrix for an undirected graph? 4. What is the significance of isolated vertices in a graph? 5. How would the adjacency matrix change if the graph was directed? **Tip**: Always double-check the elements of a set when performing union, intersection, or complement operations to avoid missing elements.