Question

For example I have sets

A={2,3,4}
B={3,4,5}
C={1,2,3}

for some reason I can't do |AUBUC|, only what i can do is calculate |A|, |B|, |C|.

How do I do to calculate |AUBUC| in this situation? is there any algorithm?

Answer 1

To extend the last sentence of Ilya's reply have a look in Wikipedia for the inclusion-exclusion principle which accounts for the size of the intersections by the formula

$$ \left| A \cup B \cup C \right | = \left| A \right| + \left| B \right| + \left| C \right| - \left| A \cap B\right| - \left| B \cap C \right| - \left| A \cap C \right| + \left| A \cap B \cap C\right|. $$

There is an obvious generalisation to an alternating sum over the cardinalities of all the $k$-fold intersections in the case of $n$ sets. Of course if you don't know the cardinalities of the intersections this is not so useful!

Answer 2

I do not know, how much is it related to set-theory, but from the measure-theoretical point of view, $|A| = \mu(A)$ where $\mu$ is a counting measure, i.e. the one which assigns the unit weight to each element of the set. In general, if you have a finite number of sets $A_1,\dots,A_n$ then $$ \mu(A_1\cup\dots\cup A_n)\leq\sum\limits_{i=1}^n\mu( A_i) $$ which is quite a trivial fact in your case. The strict inequality only holds when the sets are not disjoint, i.e. $A_i\cap A_j$ is not empty for some $i\neq j$. In that case, you have also to account for how many element are in the intersection - and to be able to compute $\mu(A_1\cap A_2)$, $\mu(A_1\cap A_2\cap A_3)$ etc.