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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:
\[
\begin{bmatrix}
0 & 1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]

#### **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:
\[
\begin{bmatrix}
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 0
\end{bmatrix}
\]

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Would you like me to explain any specific part in more detail?

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### **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.

Explore detailed solutions for verifying set theory properties, function compositions, and understanding adjacency matrices in graph theory. This covers complement operations, union, intersection, and function composition with practical examples and step-by-step solutions. Suitable for Grades 10-12 students.

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