In the last several lessons we’ve seen many ways to take two sets and form a new set. The four constructions that we studied were the union, the intersection, the Cartesian product, and the disjoint union. In other words, these are four mathematical structures that we’ve built from the more fundamental structure of sets. There are, of course, several other structures that we could construct, however. I challenge you to come up with some definitions of your own for bringing together two sets to form a new one. Feel free to tell me and the rest of the world about them in the comments section, and don’t be shy!
Once you’ve found one, you’ll have built your very own mathematical structure! All you need to do is make sure the construction is well-defined as a set and we’re good. From there, you can ask questions about it—poke it, prod it, examine it, all with the sharpened blade of logic!
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The True Beauty of Math 
The disjoint intersection (DI*) of A and B labels each element in the intersection of A and B, so if
A = {1, 2, 3} and B = {2, 3, 4}, then A DI B = { (2, A), (3, A), (2, B), (3, B) }.
Couldn’t for the life of me find an upside down disjoint union, so just picture that.
Sweet! I’ve never thought of that before, nor even heard it mentioned, but that’s a cool construction. This is interesting, because it always has twice as many elements as the normal intersection, because we first do the regular intersection and then make a “second copy” of each element to be able to label it properly. This isn’t the same for the disjoint union, which is in many ways completely independent of the regular union. For example, the DI of two sets whose regular intersection is empty is itself empty (after all,
. There is no similar relation between disjoint unions and regular unions for the number of elements in each.
Thanks for sharing!