I have a total of m elements distributed in N sets, each represented by 1 to k < m elements. My goal is to choose the minimum number of elements needed to uniquely identify all N sets.
For example: suppose there are 4 sets (A, B, C, D) containing 6 elements (1, 2, 3, 4, 5, 6)
A: {1, 3, 5, 6}
B: {2, 4, 5, 6}
C: {1, 2, 4, 6}
D: {1, 4, 5}
checking each set for 3 would yield truth values of 1, 0, 0 ,0 for sets A, B, C, D
checking each set for 2 would yield truth values of 0, 1, 1, 0 for sets A, B, C, D
checking each set for 1 would yield truth values of 1, 0, 1, 1 for sets A, B, C, D
thus after checking 3 elements, each set is represented by a unique truth table (A: {1, 0, 1}; B: {0, 1, 0,} etc.). The minimum number of elements needed is thus 3, and the elements are 1, 2, and 3. How would i generalize this approach to m elements and N sets?
Let $A_{N\times m} = (a)_{i,j}$ be a binary matrix where $j$ belongs to the set $i$ if $a_{i,j}=1$. This gives a bijection between the binary matrices and your instances.
Then your problem is equivalent to find the smallest subset of columns that keeps each row distinct. This problem is known as the "Distinct vectors problem", and is known to be $\mathcal{NP}$-Hard, even for binary matrices.