The number of subsets of the first set is 30 more than the second. Find the number of elements in each set.

Let the first set have $M$ elements. The number of subsets for the first set is $2^M$.
Let the second set have $N$ elements. The number of subsets for the second set is $2^N$.

The number of subsets of the first set is 30 more than the second:

Factor out $2^N$:

We need to find factors of 30 that are powers of 2. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
The only power of 2 that is a factor of 30 is $2^1 = 2$.
So, we must have $2^N = 2$.
Therefore, $N=1$.

Divide both sides by 2:

Add 1 to both sides:

Express 16 as a power of 2:

Equate the exponents:

The values are $M=5$ and $N=1$.

The final answer is $M=5, N=1$.