Question about duplicate subsets in a partition. Similar to the example @ time=5:25, if you define the set S such that it's the set of all integers Z, and let set A1 to even integers let set A2 to all odd integers let set A3 to even integers Part 1 of the definition would have us check the union of all sets A1, A2, and A3, which would be Z (duplicates in A1 and A3 be discarded right?) So that seems to check out. Part 2, Ai, and Aj would either be equal, or disjoint, so looking at our partitions... A1 disjoint A2 A1 = A3 A2 disjoint A3 along with A1=A1, A2=A2, A3=A3 (when i=j) which appears to satisfy the conditions in part 2. So does this imply that you can add additional subsets to any already defined partition, as long as it's duplicating another partition? It seems that the definition of a partition does not have a uniqueness requirement among subsets, and you could have an infinite number of subsets in any partition. -Also, thanks for the videos!!!
At 4:50... is it not easier to say that (i != j)? Why entertain the idea of two equal sets within a partition? It makes me think that there can be infinitely many cells (other than just A_j) that are also equal to A_i. I would imagine that there is going to be a need to count the # of cells in a partition, and this does not seem to set that up well. A cleaner way might be to use a double arbitrary unions of paired intersections equaling the empty set, such as U_{i}^{j}(U_{j}^{# of cells}(A_i intersect A_j)) = { }. Obviously not easy to read this way, but when using LaTeX of handwriting, it feels less ambiguous.
