# Help with set property

### Help with set property

I'm considering these data in my set:

https://i.stack.imgur.com/mhKUq.png

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

$$\operatorname{Part}(A) = \{\{1, 2\}, \{3, 4, 5\}, \{6, 7, 8\}\}$$

$$\operatorname{Part}(B) = \{\{1\}, \{2, 3, 4\}, \{5, 6\}, \{7,8\}\}$$

$$\operatorname{Part}(C) = \{\{1, 3, 4, 6\}, \{2, 5, 7\}, \{8\}\}$$

$$\operatorname{Part}(D) = \{\{1, 4, 7\}, \{2\}, \{3\}, \{5\}, \{6\}, \{8\}\}$$

If I consider multiple column, I'll have:

$$\operatorname{Part}(AB) = \{\{3, 4\}, \{1\}, \{2\}, \{5\}, \{6\}, \{7, 8\}\}$$
$$\operatorname{Part}(ABC) = \{\{3, 4\}, \{1\}, \{2\}, \{5\}, \{6\}, \{7\}, \{8\}\}$$
$$\operatorname{Part}(ABCD) = \{\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}, \{8\}\}$$

I've this formula that calculate the error in a partition:

$$err(Part(X)) = ||Part(X)|| - |Part(X)|$$

where:

- $$||Part(X)||$$ is the total number of element in a set (in the example
is
- $$|Part(X)|$$ 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:

$$\operatorname{err(Part(A))} = 8 - 3 = 5$$

$$\operatorname{err(Part(B))} = 8 - 4 = 4$$

$$\operatorname{err(Part(C))} = 8 - 3 = 5$$

$$\operatorname{err(Part(D))} = 8 - 6 = 2$$

$$\operatorname{err(Part(AB))} = 8 - 6 = 2$$

$$\operatorname{err(Part(ABC))} = 8 - 7 = 1$$

$$\operatorname{err(Part(ABCD))} = 8 - 8 = 0$$

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:

$$\operatorname{err(Part(A))} = 8 - 3 = 5$$

$$\operatorname{err(Part(B))} = 8 - 4 = 4$$

And I know the total number of elements (i.e. :

$$\operatorname{err(Part(AB))} = 8 - f(x) = 2$$

$$\operatorname{err(Part(ABC))} = 8 - f(x_1) = 1$$

$$\operatorname{err(Part(ABCD))} = 8 - f(x_2) = 0$$

Can I calculate $$f(x)$$, $$f(x_1)$$, $$f(x_2)$$ without knowing the AB, ABC and ABCD partitions, respectively?
cciro94

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Joined: Fri Jun 28, 2019 1:56 pm
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