Set Partitions with refinement property

[stefanaimon]

I'm considering these data in my set:


for each attribute A,B,C, and D I've these partitions:

[MATH]\operatorname{Part}(A) = \{\{1, 2\}, \{3, 4, 5\}, \{6, 7, 8\}\}[/MATH]
[MATH]\operatorname{Part}(B) = \{\{1\}, \{2, 3, 4\}, \{5, 6\}, \{7,8\}\}[/MATH]
[MATH]\operatorname{Part}(C) = \{\{1, 3, 4, 6\}, \{2, 5, 7\}, \{8\}\}[/MATH]
[MATH]\operatorname{Part}(D) = \{\{1, 4, 7\}, \{2\}, \{3\}, \{5\}, \{6\}, \{8\}\}[/MATH]
If I consider multiple column, I'll have:

[MATH]\operatorname{Part}(AB) = \{\{3, 4\}, \{1\}, \{2\}, \{5\}, \{6\}, \{7, 8\}\}[/MATH][MATH]\operatorname{Part}(ABC) = \{\{3, 4\}, \{1\}, \{2\}, \{5\}, \{6\}, \{7\}, \{8\}\}[/MATH][MATH]\operatorname{Part}(ABCD) = \{\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}, \{8\}\}[/MATH]
I've this formula that calculate the error in a partition:

[MATH]err(Part(X)) = ||Part(X)|| - |Part(X)|[/MATH]
where:

- [MATH]||Part(X)||[/MATH] is the total number of element in a set (in the example
is 8)
- [MATH]|Part(X)|[/MATH] is the number of subset in each partition (for
example in part(A) is 3 and in part(B) is 4)

and in example the error values are:

[MATH]\operatorname{err(Part(A))} = 8 - 3 = 5[/MATH]
[MATH]\operatorname{err(Part(B))} = 8 - 4 = 4[/MATH]
[MATH]\operatorname{err(Part(C))} = 8 - 3 = 5[/MATH]
[MATH]\operatorname{err(Part(D))} = 8 - 6 = 2[/MATH]
[MATH]\operatorname{err(Part(AB))} = 8 - 6 = 2[/MATH]
[MATH]\operatorname{err(Part(ABC))} = 8 - 7 = 1[/MATH]
[MATH]\operatorname{err(Part(ABCD))} = 8 - 8 = 0[/MATH]
Is there a way I would to calculate the error of multiple column partitions starting from the error of the single partition column?

For example if I've:

[MATH]\operatorname{err(Part(A))} = 8 - 3 = 5[/MATH]
[MATH]\operatorname{err(Part(B))} = 8 - 4 = 4[/MATH]
And I know the total number of elements (i.e. 8):

[MATH]\operatorname{err(Part(AB))} = 8 - f(x) = 2[/MATH]
[MATH]\operatorname{err(Part(ABC))} = 8 - f(x_1) = 1[/MATH]
[MATH]\operatorname{err(Part(ABCD))} = 8 - f(x_2) = 0[/MATH]
Can I calculate [MATH]f(x)[/MATH], [MATH]f(x_1)[/MATH], [MATH]f(x_2)[/MATH] without knowing the AB, ABC and ABCD partitions, respectively?