The following matrix is 2-separable, because each pair of columns has a distinct sum. For example, the boolean sum (that is, the bitwise OR) of the first two columns is ; that sum is not attainable as the sum of any other pair of columns in the matrix.
However, this matrix is not 3-separable, because the sum of columns 1, 2, and 3 (namely ) equals the sum of columns 1, 4, and 5.
This matrix is also not -separable, because the sum of columns 1 and 8 (namely ) equals the sum of column 1 alone. In fact, no matrix with an all-zero column can possibly be -separable for any .
The following matrix is -separable (and thus 2-disjunct) but not 3-disjunct.
There are 15 possible ways to choose 3-or-fewer columns from this matrix, and each choice leads to a different boolean sum:
More information columns, boolean sum ...
columns |
boolean sum |
|
columns |
boolean sum |
none |
000000 |
|
2,3 |
011110 |
1 |
110000 |
|
2,4 |
101101 |
2 |
001100 |
|
3,4 |
111011 |
3 |
011010 |
|
1,2,3 |
111110 |
4 |
100001 |
|
1,2,4 |
111101 |
1,2 |
111100 |
|
1,3,4 |
111011 |
1,3 |
111010 |
|
2,3,4 |
111111 |
1,4 |
110001 |
|
|
|
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However, the sum of columns 2, 3, and 4 (namely ) is a superset of column 1 (namely ), which means that this matrix is not 3-disjunct.