The number of subsets of a set with $mm$m elements is $2^{m}2^m$2m, and for a set with $nn$n elements, it is $2^{n}2^n$2n. Given the relationship:

$2^m=2^n+562^m\; =\; 2^n\; +\; 56$2m=2n+56

To find integer values for $mm$m and $nn$n, we can test possible values:

1. If $n=5n\; =\; 5$n=5: $2^n=322^n\; =\; 32$2n=32

   $2^m=32+56=88(not a power of 2)2^m\; =\; 32\; +\; 56\; =\; 88\; \backslash quad\; \backslash text\{(not\; a\; power\; of\; 2)\}$2m=32+56=88(not a power of 2)

2. If $n=6n\; =\; 6$n=6: $2^n=642^n\; =\; 64$2n=64

   $2^m=64+56=120(not a power of 2)2^m\; =\; 64\; +\; 56\; =\; 120\; \backslash quad\; \backslash text\{(not\; a\; power\; of\; 2)\}$2m=64+56=120(not a power of 2)

3. If $n=7n\; =\; 7$n=7: $2^n=1282^n\; =\; 128$2n=128

   $2^m=128+56=184(not a power of 2)2^m\; =\; 128\; +\; 56\; =\; 184\; \backslash quad\; \backslash text\{(not\; a\; power\; of\; 2)\}$2m=128+56=184(not a power of 2)

4. If $n=4n\; =\; 4$n=4: $2^n=162^n\; =\; 16$2n=16

   $2^m=16+56=72(not a power of 2)2^m\; =\; 16\; +\; 56\; =\; 72\; \backslash quad\; \backslash text\{(not\; a\; power\; of\; 2)\}$2m=16+56=72(not a power of 2)

5. If $n=3n\; =\; 3$n=3: $2^n=82^n\; =\; 8$2n=8

   $2^m=8+56=64\Rightarrow m=62^m\; =\; 8\; +\; 56\; =\; 64\; \backslash quad\; \backslash Rightarrow\; m\; =\; 6$2m=8+56=64⇒m=6

Thus, $m=6m\; =\; 6$m=6 and $n=3n\; =\; 3$n=3. Therefore, $m+n=6+3=9m+n\; =\; 6\; +\; 3\; =\; 9$m+n=6+3=9.