What is a set of sets called?

So, What Do You Call a Bunch of Sets? Let’s Untangle This!

Okay, so you know what a set is, right? It’s basically a collection of stuff – numbers, letters, whatever. But things get interesting when the “stuff” inside the set is… more sets! This tripped me up for a while when I first encountered it. So, what do you even call that?

Well, good news: there isn’t some secret, super-official term that’ll get you kicked out of the math club if you don’t know it. But there are a few common ways to describe it, and understanding them can make wading through set theory a whole lot easier.

The most obvious? “A set of sets.” Yep, sometimes the simplest answer is the best. It’s clear, it’s straightforward, and everyone knows what you’re talking about.

But you’ll also hear people say “a family of sets.” I kind of like this one; it makes it sound like all the little sets are related, hanging out together. Often, “family” does imply they have something in common – maybe they all contain even numbers, or they’re all subsets of a bigger set. Think of it like a family reunion, but with curly braces instead of awkward small talk.

“A collection of sets” is another option. It’s pretty similar to “a set of sets,” maybe just a touch less formal. Think of it like this: “set” sounds like you carefully curated the group, while “collection” just implies you gathered them together.

Now, you might stumble across the term “class of sets.” Be a little careful with this one. In advanced math, “class” can have a very specific meaning that’s different from “set.” So, while it’s not wrong, it could raise some eyebrows in certain circles.

Let’s make this concrete with an example. Say I have set A, which is {1, 2, 3}, and set B, which is {4, 5, 6}. If I create a new set S that contains A and B, then S = {{1, 2, 3}, {4, 5, 6}}. Boom! That’s a set of sets. S has two things in it, and each of those things is, itself, a set.

Here’s another cool example: the power set. The power set of any set is always a set of sets. Remember those from your discrete math class? If you start with a set – let’s say {a, b} – its power set is all the possible subsets you can make from it: {}, {a}, {b}, and {a, b}. Put them all together, and you get { {}, {a}, {b}, {a, b} } – a set containing other sets.

Why does any of this matter? Well, being able to talk about sets of sets lets us build more complicated stuff in math. It’s like having LEGOs instead of just plain blocks. You can connect things in new ways and create more interesting structures. Set theory, which got its start with the work of Georg Cantor, is super important for how we understand math today.

And it doesn’t stop there! You can have sets of sets of sets! It’s sets all the way down (or up, depending on how you look at it). While there isn’t a super common term for that, you could reasonably call it “a set of families of sets” to keep things clear. In the really deep end of set theory, they use something called the “rank” of a set to keep track of these nested levels.

So, the next time you’re wrestling with sets and subsets, remember that a “set of sets,” a “family of sets,” or a “collection of sets” are all perfectly good ways to describe a set whose elements are, themselves, sets. Now you can confidently untangle those curly braces and conquer the world of set theory!