Answer to Question #148124 in Discrete Mathematics for Promise Omiponle

Question #148124
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.
1
Expert's answer
2020-12-16T20:04:24-0500

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


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


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