Question

Let C={A1, A2, ..., An} be a collection of finite sets that are pairwise disjoint. Further suppose that |Ai|=i. Compute |U(i=1 to n)Ai|, and write your answer in the simplest closed form possible.

Answer 1

$C=\{A_1,A_2,\cdots, A_n\}, |A_i|=i\\ \left| \bigcup_{i=1}^nA_i\right|=\left|A_1 \bigcup A_2 \bigcup \cdots \bigcup A_n\right|\\ =|A_1|+|A_2|+ \cdots + |A_n|\\ \text{Since they are disjoint}\\ =1+2+ \cdots +n\\ = \sum_{i=1}^ni\\ =\frac{n(n+1)}{2}$

Hence, $\left| \bigcup_{i=1}^nA_i\right|=\frac{n(n+1)}{2}.$