![]() Let us consider the following two sets: and. Our first example will seem kind of tedious and unnecessary, but it will truly illustrate the power of the idea of “sets of sets”, and it will hopefully make clearer the distinction between elements, sets, subsets, and sets of sets. Therefore we are perfectly able to consider a set whose individual, indivisible elements are entire sets. After all, a set is a “thing”, and we have a well-defined notion of when two sets are distinct from each other. Accordingly, we can think of a set itself as a single element of some other set. As discussed before, elements could be numbers, or donkeys, or ideas, or some combination of any of these and anything else that you can think of. ![]() Namely, all we need is a collection of distinct “things” which we call elements. We begin by recalling what we need to make a set. Okay, I know this might seem a bit strange and a bit too abstract, so let’s take this slowly (because it’s extremely important). So what is a set of sets? As usual, it is exactly what it sounds like: a set whose elements are themselves sets. After exploring this idea for a bit, we’ll see where the trouble comes from in this lesson and the next one. In other words, the idea of “sets of sets” is not only extremely useful, but also extremely problematic (at times). We now go on to study the notion of “sets of sets”, which not only is an extremely powerful mathematical tool, but also is the tip of an incredibly difficult mathematical iceberg. In lesson 2 we defined and discussed sets, and in lesson 3 we studied subsets of those sets.
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