EDB — 060

Exercises

1. [060] Let \(X\) be a set. Let \(I,J\) families not empty of indexes, and for every \(i∈ I\) let \(J_ i⊆ J\) a family not empty of indexes. For each \(i∈ I, j∈ I_ j\) let \(A_{i,j}⊆ X\). Show that

   \[ ⋂_{i∈ I}⋃_{j∈ J_ i}A_{i,j}= ⋃_{𝛽∈ B} ⋂_{i∈ I} A_{i,𝛽(i)} \]

   where \(B=∏_{i∈ I} J_ i\) and remember that every \(𝛽∈ B\) is a function \(𝛽:I→ J\) for which for every \(i\) you have \(𝛽(i)∈ J_ i\). Then formulate a similar rule by exchanging the role of intersection and union. (use the complements of the sets \(A_{i,j}\) and the rules of de Morgan).

   Solution 1

   [061]↺↻

   [[27B]↺↻]