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 sum 0 by convention
To determine the number of subsets of {1, 2, 3, 4, 5, 6, 7, 8} that have distinct subset sums, we can use a brute force approach. Here's how you can do it:
1. Start by generating all possible subsets of the given set. Since the set has 8 elements, there will be 2^8 = 256 possible subsets.
2. For each subset, calculate the sum of its elements.
3. Keep track of all the distinct sums you encounter.
4. Finally, count the number of distinct sums you found.
Here's an example step-by-step process:
1. Generate all possible subsets of {1, 2, 3, 4, 5, 6, 7, 8}:
- Subset 1: {}
- Subset 2: {1}
- Subset 3: {2}
- Subset 4: {1, 2}
- ...
- Subset 256: {1, 2, 3, 4, 5, 6, 7, 8}
2. Calculate the sum of each subset:
- Subset 1: sum = 0
- Subset 2: sum = 1
- Subset 3: sum = 2
- Subset 4: sum = 3
- ...
- Subset 256: sum = 36
3. Keep track of all distinct sums encountered:
- Distinct sums: {0, 1, 2, 3, ..., 36}
4. Finally, count the number of distinct sums: 37
Therefore, there are 37 subsets of {1, 2, 3, 4, 5, 6, 7, 8} that have distinct subset sums.
Note: This brute force method works well for small sets, but it becomes computationally expensive for larger sets. In those cases, more efficient algorithms, such as dynamic programming or recursive approaches, can be used to solve the problem efficiently.