How To Prove It by Velleman

Prove It by Velleman, Section 2.3, Problem 9

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Problem

Give an example of an index set I and indexed families of set { A_i \mid i \in I } and { B_i \mid i \in I } such that \bigcup \limits_{i \in I} (A_i \cap B_i) \neq (\bigcup \limits_{i \in I} A_i) \cap (\bigcup \limits_{i \in I} B_i) .

Solution

The trick here is to make sure that A_j and B_i have overlap but not overlap A_i , i \neq j . These overlaps would be excluded from \bigcup \limits_{i \in I} (A_i \cap B_i) (because it’s not in B_i ) but still be in (\bigcup \limits_{i \in I} A_i) \cap (\bigcup \limits_{i \in I} B_i) .

I = {1, 2}

A_i = { i, i + 1 }

B_i = { i, i + 2 }

A_1 = { 1, 2 }

A_2 = { 2, 3 }

B_1 = { 1, 3 }

B_2 = { 2, 4 }

A_1 \cap B_1 = { 1 }

A_2 \cap B_2 = { 2 }

\bigcup \limits_{i \in I} A_i = { 1, 2, 3 }

\bigcup \limits_{i \in I} B_i = { 2, 3, 4 }

\bigcup \limits_{i \in I} (A_i \cap B_i) = { 1, 2 }

(\bigcup \limits_{i \in I} A_i) \cap (\bigcup \limits_{i \in I} B_i) = { 1, 2, 3, 4 }

{1, 2 } \neq { 1, 2, 3, 4 }

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