# Calculate cardinality of a set

There is given:

• set $$B=A_{1} \cup...\cup A_{n}$$
• $$|A_{i}|=m_{i}$$ for each $$i$$
• every element of $$B$$ belongs to exactly $$k$$ sets $$A_{i_{1}},...,A_{i_{k}}$$

Calculate $$|B|$$ in terms of $$m_{i}$$. If it has not unique solution then find its upper bound.

I tried to use inclusion-exlusion principle but so far my estimation is too big. It is $$m_{1}+...+m_{n}$$

Is there any better estimations?

Regards.

• Hint: If you union all of them you will have $k$ copies of every element. – G Aker Apr 24 at 9:20

Every element of $$B$$ belongs to exactly $$k$$ sets out of the $$A_i$$. So if we count all the elements of the $$A_i$$, we have counted each element of $$B$$ exactly $$k$$ times. This means that $$|A_1|+|A_2|+\cdots+|A_n|=k|B|.$$