EDB — 060

Exercises

1. [060] Let $X$ be a set. Let $I,J$ families not empty of indexes, and for every $i\in I$ let $J_i\subseteq J$ a family not empty of indexes. For each $i\in I,j\in I_j$ let $A_{i,j}\subseteq X$. Show that

   $${\bigcap }_{i\in I}{\bigcup }_{j\in J_i}{A}_{i,j}={\bigcup }_{𝛽\in B}{\bigcap }_{i\in I}{A}_{i,𝛽(i)}$$

   where $B={\prod }_{i\in I}{J}_i$ and remember that every $𝛽\in B$ is a function $𝛽:I\to J$ for which for every $i$ you have $𝛽(i)\in J_i$. 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]↺↻]