N-uniquely additive sets

Jelly, 7 bytes


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Returns a positive number for truthy and zero for falsey.

œc       find combinations
  S€     sum each combination
    Ṣ    sort the sums
     I   find the difference between each pair of sums 
           iff any sums are the same, this returns a list containing 0
      P  product of the elements of the resulting list

Matlab, 78 bytes

function n=f(s,n);p=perms(s);k=sum(unique(sort(p(:,1:n)),'rows')');unique(k)-k

This function returns a positive value (in fact n) for truthy and returns an error as a falsey answer (valid according to this comment)


function n=f(s,n);
p=perms(s); %create all permutations of the set

                  %just take the first n entries of each permutation
             %sort those entries and
      %filter out all duplicates (we sorted as the order should NOT matter)
  %then sum each of those candidates

%if all those sums are distinct, unique(k) will have the same size 
% as k itself, and therefore we can subtract, otherwise it will throw 
% an error as we try to subtract vectors of different sizes

MATL, 7 bytes


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Returns true (displayed as 1) or false (displayed as 0).

XN   % Take array S and number N. Generate all combinations of elements from S 
     % taken N at a time. Gives a 2D array where each combination is a row
!    % Transpose. Each combination is now a column
s    % Sum of each column: gives a row array. If N=1 computes the sum of
     % the only row, and so gives a number
S    % Sort vector
d    % Array of consecutive differences. For a single number gives an empty array
A    % True if all elements of the input array are nonzero (for an empty array
     % it also gives true)