Can I interchange union and intersection over indexed sets this way?

[wayneckm]

Hello all,


I have the following question regarding the interchange between union and intersection.

$$\cup_{q < t} \cap_{s > q} A_{s} = \cap_{s<t} \cup_{q<s} A_{q} = \cup_{q < t} A_{q}$$

Am I correct? Also, can anyone provide me some more resources regarding this kind of interchange in set operation or change of variable technique?

Thanks.


Wayne

 

[yossell]

Off the top of my head, it doesn't look correct.

Suppose Ai is empty for all i>100. Then $$\cap_{s>q}A_s$$ will be empty for all q, and the union of these will be empty. So lhs is empty.

But rhs, being just a union of As, will not necessarily be empty.