For a set of numbers T, we say that T has distinct subset sums if all distinct subsets of T have distinct sums. How many subsets of {1,2,3,4,5,6,7,8} have distinct subset sums?
Details and assumptions
The empty set (the set of no elements) has a sum of 0 by convention.
To find out how many subsets of {1, 2, 3, 4, 5, 6, 7, 8} have distinct subset sums, we can use a bitset approach.
1. First, we convert each number in the set to a binary representation.
- {1, 3} becomes {001, 011}
- {2, 5} becomes {010, 101}
- {3, 8} becomes {011, 1000}
- ...
- {8} becomes {1000}
2. Next, for each binary representation, we create a binary string that represents the sum of the corresponding elements.
- {001, 011} becomes {000, 001, 010, 011}
- {010, 101} becomes {000, 010, 100, 110, 101, 111}
- {011, 1000} becomes {000, 011, 100, 111, 1000, 1011, 1100, 1111}
3. Finally, we count the number of unique binary strings from step 2.
- {000, 001, 010, 011, 100, 111, 1100, 1111} has 8 unique subset sums.
Therefore, there are 8 subsets of {1, 2, 3, 4, 5, 6, 7, 8} that have distinct subset sums.