Two infinite sets have `m` and `n` elements. The number of subsets of the first set is `112` more than that of the second set. The values of `m` and `n` are, respectively.(a) `4, 7`(b) 7,4`(c) `4,4`(d) 7,7`

Options

A. `4, 7`
B. `7,4`
C. `4,4`
D. `7,7`

Explanation

To solve the problem, we need to determine the values of \( m \) and \( n \) based on the given information about the number of subsets of two sets.

### Step 1: Understand the formula for the number of subsets
The number of subsets of a set with \( k \) elements is given by \( 2^k \). Therefore, for the first set with \( m \) elements, the number of subsets is \( 2^m \), and for the second set with \( n \) elements, the number of subsets is \( 2^n \).

**Hint:** Recall that the number of subsets of a set is calculated using the formula \( 2^k \) where \( k \) is the number of elements in the set.

### Step 2: Set up the equation based on the problem statement
According to the problem, the number of subsets of the first set is 112 more than that of the second set. This can be expressed mathematically as:
\[ 2^m = 2^n + 112 \]

**Hint:** Write down the equation based on the relationship given in the problem.

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ 2^m - 2^n = 112 \]

**Hint:** Isolate \( 2^m \) on one side to clearly see the relationship between \( m \) and \( n \).

### Step 4: Factor the left-hand side
We can factor the left-hand side:
\[ 2^n(2^{m-n} - 1) = 112 \]

**Hint:** Factor out \( 2^n \) to simplify the equation.

### Step 5: Express 112 in terms of powers of 2
We can express 112 as:
\[ 112 = 16 \times 7 = 2^4 \times 7 \]

**Hint:** Break down 112 into its prime factors to see how it relates to powers of 2.

### Step 6: Analyze the equation
From the equation \( 2^n(2^{m-n} - 1) = 2^4 \times 7 \), we can deduce that \( 2^n \) must be a power of 2. The possible values for \( 2^n \) that are less than or equal to 112 are \( 1, 2, 4, 8, 16 \).

**Hint:** Consider the possible values of \( n \) based on the powers of 2.

### Step 7: Test possible values for \( n \)
- If \( n = 4 \), then \( 2^n = 16 \):
\[ 16(2^{m-4} - 1) = 112 \implies 2^{m-4} - 1 = 7 \implies 2^{m-4} = 8 \implies m - 4 = 3 \implies m = 7 \]

Thus, we find \( m = 7 \) and \( n = 4 \).

**Hint:** Substitute back to check if your values satisfy the original equation.

### Conclusion
The values of \( m \) and \( n \) are \( 7 \) and \( 4 \), respectively.

### Final Answer
The values of \( m \) and \( n \) are \( 7 \) and \( 4 \).