If `A=(2,3,4,8,10},B=(3,4,5,10,12}andC=(4,5,6,12,14),` find `(AuuB)nn(AuuC)and(AnnB)uu(AnnC)`.

Answer

To solve the problem, we need to find two expressions involving the sets A, B, and C. The sets are defined as follows:

- \( A = \{2, 3, 4, 8, 10\} \)
- \( B = \{3, 4, 5, 10, 12\} \)
- \( C = \{4, 5, 6, 12, 14\} \)

We need to find:

1. \( (A \cup B) \cap (A \cup C) \)
2. \( (A \cap B) \cup (A \cap C) \)

### Step 1: Calculate \( A \cup B \)

The union of sets A and B includes all unique elements from both sets.

- Elements in A: \( 2, 3, 4, 8, 10 \)
- Elements in B: \( 3, 4, 5, 10, 12 \)

Combining these, we get:
\[ A \cup B = \{2, 3, 4, 5, 8, 10, 12\} \]

### Step 2: Calculate \( A \cup C \)

Now, we find the union of sets A and C.

- Elements in A: \( 2, 3, 4, 8, 10 \)
- Elements in C: \( 4, 5, 6, 12, 14 \)

Combining these, we get:
\[ A \cup C = \{2, 3, 4, 5, 6, 8, 10, 12, 14\} \]

### Step 3: Calculate \( (A \cup B) \cap (A \cup C) \)

Now we find the intersection of \( A \cup B \) and \( A \cup C \).

- \( A \cup B = \{2, 3, 4, 5, 8, 10, 12\} \)
- \( A \cup C = \{2, 3, 4, 5, 6, 8, 10, 12, 14\} \)

The common elements are:
\[ (A \cup B) \cap (A \cup C) = \{2, 3, 4, 5, 8, 10, 12\} \cap \{2, 3, 4, 5, 6, 8, 10, 12, 14\} = \{2, 3, 4, 5, 8, 10, 12\} \]

### Step 4: Calculate \( A \cap B \)

Now we find the intersection of sets A and B.

- Elements in A: \( 2, 3, 4, 8, 10 \)
- Elements in B: \( 3, 4, 5, 10, 12 \)

The common elements are:
\[ A \cap B = \{3, 4, 10\} \]

### Step 5: Calculate \( A \cap C \)

Next, we find the intersection of sets A and C.

- Elements in A: \( 2, 3, 4, 8, 10 \)
- Elements in C: \( 4, 5, 6, 12, 14 \)

The common element is:
\[ A \cap C = \{4\} \]

### Step 6: Calculate \( (A \cap B) \cup (A \cap C) \)

Now we find the union of \( A \cap B \) and \( A \cap C \).

- \( A \cap B = \{3, 4, 10\} \)
- \( A \cap C = \{4\} \)

Combining these, we get:
\[ (A \cap B) \cup (A \cap C) = \{3, 4, 10\} \cup \{4\} = \{3, 4, 10\} \]

### Final Answers

1. \( (A \cup B) \cap (A \cup C) = \{2, 3, 4, 5, 8, 10, 12\} \)
2. \( (A \cap B) \cup (A \cap C) = \{3, 4, 10\} \)