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 approach the problem using a brute force method. Here are the steps:
1. Create a list to keep track of all the subset sums.
2. Generate all possible subsets of the given set {1, 2, 3, 4, 5, 6, 7, 8}. This can be done using the "power set" concept, where each element can either be included or excluded to form subsets.
3. For each subset, calculate the sum of its elements and add it to the list of subset sums.
4. After generating all the subsets and summing their elements, check if there are any duplicate sums in the list. If there are no duplicates, then the subset is distinct.
5. Finally, count the number of distinct subsets.
Here's an example of applying this method:
1. Start with an empty list to store the subset sums.
2. Generate all possible subsets:
- Empty set: {}
- Subset with only one element: {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}
- Subset with two elements: {1, 2}, {1, 3}, {1, 4}, ..., {7, 8}
- Subset with three elements: {1, 2, 3}, {1, 2, 4}, ..., {6, 7, 8}
- Subset with four elements: {1, 2, 3, 4}, {1, 2, 3, 5}, ..., {5, 6, 7, 8}
- Subset with five elements: {1, 2, 3, 4, 5}, ..., {4, 5, 6, 7, 8}
- Subset with six elements: {1, 2, 3, 4, 5, 6}, ..., {3, 4, 5, 6, 7, 8}
- Subset with seven elements: {1, 2, 3, 4, 5, 6, 7}, ..., {2, 3, 4, 5, 6, 7, 8}
- Subset with eight elements: {1, 2, 3, 4, 5, 6, 7, 8}
3. Calculate the sum of each subset:
- Empty set has a sum of 0.
- Subset with only one element has a sum equal to the element itself.
- For subsets with more than one element, iterate through the elements and sum them.
4. Check for duplicate sums:
- Compare each sum with the previously calculated sums in the list.
- If a duplicate sum is found, discard the subset.
- If the sum is unique, add it to the list.
5. Count the number of distinct subsets by counting the number of sums in the list.
By following these steps, you should be able to determine the number of subsets of {1, 2, 3, 4, 5, 6, 7, 8} that have distinct subset sums.