Aug 11, 2020
In Mathematical Proofs
Nearly all of the examples I can find regarding union or intersections with indexed families of sets will show some particular set B unioned/intersected with the union/intersection of Ai where 'i' is in I (I being the index of a family of sets). In these cases, the union/intersection can be distributed across the two pretty easily. So B U (∩[i<I]Ai) ends up being ∩[i<I](B U Ai) but what if B was actually Ci.. in that case, (∩[i<I]Ai) U (∩[i<I]Ci)... does the distributive property hold such that ∩[i<I](Ai U Ci)? and then following along.. when computing ∩[i<I](Ai U Ci) : why does it seem that we compute the unions of the indexed sets (Ai U Aj) and (Bi U Bj) [where i and j are both in I] before we compute the intersection between the two? if we work back from ∩[i<I](Ai U Ci) -> (∩[i<I]Ai) U (∩[i<I]Ci) [presuming my intuition here is right), shouldn't we perform the intersection of the indexed families and then union the results together?