at set is $$2^n$$.

According to the problem, the total number of subsets of the first set is 112 more than the total number of subsets of the second set. Therefore, we can write the equation:

$$2^m = 2^n + 112$$

We need to find the values of $$m$$ and $$n$$ that satisfy this equation. Let's check the given options one by one:

Option A: $$m = 7, n = 4$$

Substitute these values into the equation:

$$2^7 = 2^4 + 112$$

Calculate the powers of 2:

$$128 = 16 + 112$$

This simplifies to:

$$128 = 128$$

This is a true statement, so Option A is correct.

Option B: $$m = 7, n = 7$$

Substitute these values into the equation:

$$2^7 = 2^7 + 112$$

Calculate the powers of 2:

$$128 = 128 + 112$$

This simplifies to:

$$128 = 240$$

This is not true, so Option B is incorrect.

Option C: $$m = 4, n = 4$$

Substitute these values into the equation:

$$2^4 = 2^4 + 112$$

Calculate the powers of 2:

$$16 = 16 + 112$$

This simplifies to:

$$16 = 128$$

This is not true, so Option C is incorrect.

Option D: $$m = 4, n = 7$$

Substitute these values into the equation:

$$2^4 = 2^7 + 112$$

Calculate the powers of 2:

$$16 = 128 + 112$$

This simplifies to:

$$16 = 240$$

This is not true, so Option D is incorrect.

Therefore, the correct answer is Option A: $$m = 7, n = 4$$.

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