What I'm looking for is the name of a type of number set. Given a number T (for total) and a set of positive integers S, I want to uniquely identify the subset of S that sums to T. All sets containing 1 or 2 positive integers will pass the test, since there is a unique combination of those numbers and the sum will therefore be the same.
For example:
{1, 7, 89} passes the test. Any combination of those numbers, when summed, will generate a unique T, and vice versa, any T that is a sum of a combination of those numbers will generate a unique subset of S. So, the set of all T s for the above set is {0, 1, 7, 8, 89, 90, 96, 97}.
{2, 3, 7, 8} does not pass the test. There are multiple combinations that yield a total of 10 ({2, 8}, and {3, 7}). So if I specified a T of 10, you could not tell me with confidence the combination that produced that sum.
With that out of the way. My question is this... Is there name for a set of positive integers like this? I'd like to learn more about them more a personal project of mine.
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$\begingroup$They are called sets with distinct subset sums.
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