If `A={1,2,3,4}" and "B={2,4,6}`. Then, the number of set C such that `A nn B sube C sube A uu B` is

Options

A. 6
B. 9
C. 8
D. 10

Explanation

To solve the problem, we need to find the number of sets \( C \) such that \( A \cap B \subseteq C \subseteq A \cup B \).

### Step 1: Identify the sets A and B
Given:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{2, 4, 6\} \)

### Step 2: Find the intersection of A and B
The intersection \( A \cap B \) includes the elements that are common to both sets \( A \) and \( B \):
- \( A \cap B = \{2, 4\} \)

### Step 3: Find the union of A and B
The union \( A \cup B \) includes all elements that are in either set \( A \) or set \( B \):
- \( A \cup B = \{1, 2, 3, 4, 6\} \)

### Step 4: Determine the constraints for set C
We know that:
- \( A \cap B \subseteq C \) means that \( C \) must include the elements \( 2 \) and \( 4 \).
- \( C \subseteq A \cup B \) means that \( C \) can only include elements from the set \( \{1, 2, 3, 4, 6\} \).

### Step 5: Identify the remaining elements for set C
Since \( C \) must include \( 2 \) and \( 4 \), we can consider the remaining elements from \( A \cup B \):
- Remaining elements are \( \{1, 3, 6\} \).

### Step 6: Count the combinations for set C
For each of the remaining elements \( 1, 3, \) and \( 6 \), we have two choices: either to include the element in set \( C \) or not to include it. Therefore, for each of the 3 remaining elements, we have:
- \( 2 \) choices (include or exclude).

### Step 7: Calculate the total number of sets C
Since there are 3 elements (1, 3, and 6) and each can either be included or excluded, the total number of combinations is:
- Total combinations = \( 2^3 = 8 \).

### Final Answer
Thus, the number of sets \( C \) such that \( A \cap B \subseteq C \subseteq A \cup B \) is \( 8 \).

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