Talk:Intersection Distributes over Intersection/Sets of Sets

There are cases where this doesn't hold: let $A = \set {\set {1, 2}, \set {1, 3} }$ and $B = \set {\set {1, 4}, \set {1, 5} }$. Then, one side equals $\set 1$, while the other is equal to the empty set. Univalence (talk) 12:46, 3 September 2020 (UTC)


 * Interesting. I'll bend a braincell to see if we salvage anything from this -- for example, see whether the LHS is always included in the RHS. If not, no worries, nothing depends on this, it can be deleted as a bad mistake. --prime mover (talk) 13:01, 3 September 2020 (UTC)


 * Counterexample search tool says no, given $A = \set {\set x}$ and $B = \set {\set x, \O}$. The other direction seems to hold, however, might write a proof. Univalence (talk) 13:08, 3 September 2020 (UTC)


 * Yes of course, with your original example we have $A \cap B = \O$ and so $\bigcap {A \cap B} = \Bbb U$ whatever that universe $\Bbb U$ would be in whatever context $A$ and $B$ are taken from. I'm rusty. --prime mover (talk) 13:16, 3 September 2020 (UTC)


 * Interestingly, universe stuff is not required though, take $A = \set {\set {1, 2}, \set 1}$ and $B = \set {\set {1, 2} }$. I'd just turn this into a proof of the superset statement, but I don't know how/if one would change links etc. on the parent page in this case. Univalence (talk) 13:24, 3 September 2020 (UTC)


 * Make it into something sensible and the admins can sort out the page nesting details. --prime mover (talk) 13:32, 3 September 2020 (UTC)