Let A and B be two sets. The set A has 2016 more subsets than B. If `AnnB` has 3 members, then the number of members in `AuuB` is

Options

A. 10
B. 11
C. 12
D. 13

Answer

To solve the problem step by step, we will use the information given about the sets A and B, specifically focusing on their subsets and their intersection.

### Step 1: Understand the relationship between subsets and set size
The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, if set A has \( n_1 \) elements, it has \( 2^{n_1} \) subsets, and if set B has \( n_2 \) elements, it has \( 2^{n_2} \) subsets.

### Step 2: Set up the equation based on the problem statement
According to the problem, set A has 2016 more subsets than set B. This can be expressed mathematically as:
\[ 2^{n_1} = 2^{n_2} + 2016 \]

### Step 3: Simplify the equation
We can rearrange the equation:
\[ 2^{n_1} - 2^{n_2} = 2016 \]
Factoring out \( 2^{n_2} \) gives:
\[ 2^{n_2}(2^{n_1 - n_2} - 1) = 2016 \]

### Step 4: Find possible values for \( n_1 \) and \( n_2 \)
To find suitable values for \( n_1 \) and \( n_2 \), we can start by factoring 2016. The prime factorization of 2016 is:
\[ 2016 = 2^5 \times 3^2 \times 7 \]
This means \( 2^{n_2} \) must be a power of 2 that divides 2016. The possible values for \( 2^{n_2} \) are \( 1, 2, 4, 8, 16, 32 \).

### Step 5: Testing values for \( n_2 \)
1. **If \( n_2 = 5 \)**:
\[ 2^{n_2} = 32 \]
Then:
\[ 32(2^{n_1 - 5} - 1) = 2016 \implies 2^{n_1 - 5} - 1 = 63 \implies 2^{n_1 - 5} = 64 \implies n_1 - 5 = 6 \implies n_1 = 11 \]

2. **If \( n_2 = 4 \)**:
\[ 2^{n_2} = 16 \]
Then:
\[ 16(2^{n_1 - 4} - 1) = 2016 \implies 2^{n_1 - 4} - 1 = 126 \implies 2^{n_1 - 4} = 127 \implies n_1 - 4 = 7 \implies n_1 = 11 \]
This does not yield a valid integer for \( n_1 \).

### Step 6: Calculate the union of sets A and B
Now we know:
- \( n_1 = 11 \)
- \( n_2 = 5 \)
- The intersection \( |A \cap B| = 3 \)

Using the formula for the union of two sets:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Substituting the values:
\[ |A \cup B| = n_1 + n_2 - |A \cap B| = 11 + 5 - 3 = 13 \]

### Final Answer
The number of members in \( A \cup B \) is **13**.