Let us recall what we’ve done before our detour in lesson 11 into the whacky world of the infinite. We had defined sets in lesson 2 and then realized that a natural idea to consider is a way to “relate” one set to another. I.e., once we define what a set is, we want to know about what kinds of things happened when we have two sets lying around. This motivated our definition of a function in lesson 6, which gives us a precise way of speaking about “associations” between sets. Another natural question to ask about sets is what sort of “sub-structure” they have. In other words, if you hand me a set, are there meaningful ways for me to “deconstruct” it into “smaller” things? This motivated, in a natural way, the definition of a subset that we gave in lesson 3.
These two natural questions to ask about sets carry over to all mathematical structures that one can define. This gives us a very well-defined method of moving forward in mathematics. Once one structure is defined, we can ask about what happens when two (or more) of such structures are around. This will give rise to whatever the analogy of a function is with regard to this new mathematical structure. Moreover, we can also ask what kind of “sub-structures” exist within any such structure. This would be the analog of a subset. Remember that an important quality of a subset is that it is itself a set in its own right. Thus, for any mathematical structure that we define, it is important to know if it allows some kind of sub-structure that mirrors the larger structure (we’ll see more examples of this when we see groups, topological spaces, categories, and almost everything else (although these won’t be for a while)).
Another kind of “natural” question to ask about a given mathematical structure is whether or not we can “build” new structures out of given ones. In other words, if we’re handed two of these structures, can we define a new such structure from what we were given? The answer is almost always yes, and in this lesson we’ll study this question as it pertains to sets. Note that this will in some sense complete the picture of “natural” questions to ask about a given structure. (I keep “natural” in scare quotes because by “natural” I really mean “obvious” and “necessary” in the sense that we need to ask these questions if we’re to ever make progress in studying these structures.) The reason this completes the picture is that once we have defined the structure and asked all of these “natural” questions, we’ll then know how to relate two structures to each other (the analogue of the function), find smaller structures within a structure (the analogue of a subset), and find larger structures from two smaller structures (the analogue of what we’re going to do here). We’ll then know how to relate structures, decompose structures, and build up structures. Given this framework, making progress in studying these structures becomes, well, possible.
Now that we have all of this philosophizing behind us, we can get on with mathematics. We now ask, given two sets A and B, what kind of sets can we derive from A and B? I.e., how can we build a new set from the two old sets A and B? It is this idea that gives this lesson its subtitle.
There are a few definitions that we can make, but the two that we’ll study here are unions and intersections. In short, the union of two sets A and B is the set that is formed by “bringing these sets together”, and the intersection of two sets is formed by considering the parts of the sets that are “the same”. Let us make this a bit more rigorous.
Definition 16.1 Given two sets A and B, the union of A and B is the set that consists of the elements which are either in A or in B. The union of A and B is denoted by . //
Note that we should really view as a single symbol representing the set “the union of A and B”, in the exact same way that “A” is a single symbol representing whatever A is. Thus, any element in is either an element of A or B (or possibly both), and conversely any element in A or B is also an element of .
Since this is a pretty straightforward and intuitive concept, let me take this opportunity to introduce some more notation that we’ll end up using quite a bit. To denote the fact that some element, call it “a”, is in a set A, we use the symbol “”. Thus, when I write , we should read it as “a is an element of A”. That’s not so bad.
Another bit of extremely useful notation is that which we use for defining sets. Namely, let’s say I wanted to define a set A by saying that all of the elements in A satisfy some condition. For example, I could define the set of prime numbers greater than 50 and less than 50,000. Well, constantly referring to “the set of prime numbers greater than 50 and less than 50,000” is tiring, and extremely annoying to write down. Thus, we can streamline this phrase as follows: We first let be the set of positive whole numbers (this is standard notation). Then if A is the set of prime numbers greater than 50 and less than 50,000, we write
As usual, we’re using the bracket notation to surround the elements of the set, but now we’re actually describing the elements in the set within these brackets. On the left of the vertical line we write which type of objects we’re dealing with–namely, elements in . Then on the right side of the vertical line we place the desired conditions on the elements in . The vertical line should be read as “such as”, so that the whole equation reads “A is the set of elements “p” in such that “p” is prime, and is greater than 50 and less than 50,000”. We’ll get more used to this notation by seeing some more examples. Let us therefore take the example that is appropriate for this lesson.
Using this notation, we can write the union of two sets and as follows:
This reads “the union of A and B is the set of elements “a” such that “a” is in A, or “a” is in B”. With this notation in hand, I can now define the intersection of two sets very easily.
Definition 16.2 Given two sets and , the intersection (denoted ) of and is the set of elements that are in both and . Namely, .
Let me end with a couple of remarks, and save some easy examples for the exercises. My first remark is that in order to be completely precise, I really should be using either “” or “” instead of “” in the above definitions. The reason for this is that the former two symbols (which have the exact same meaning as each other–which to use is simply a matter of taste) are read “is defined to be”, so that whatever appears on the left “is defined to be” whatever appears on the right. This differs from “” because two things can be equal to each other without being “defined to be” equal to each other. For example, if and , then clearly (they’re both the one-element set ), even though they’re defined differently. This might seem a bit pedantic, but it is actually a very nice way to keep the logic within any given argument organized. We’ll see this pop up more and more as we go.
The second and last remark I’ll make is on the importance of the empty set here. Note that if A and B are sets such that at least one of them is not empty, then their union will not be empty. However, it might be the case that the intersection of two non-empty sets is itself empty. For example, the intersection of the sets
is empty, because there is no element that is in both of these sets. Thus, the fact that the empty set is itself a set is essential if we want to speak of intersections of sets in a meaningful way.
Exploring a couple of simple examples of unions and intersections will solidify these ideas pretty quickly.
1) How many elements are in the union of the sets and ? How many are in their intersection?
2) What is the union of the sets and ? What is their intersection?
3) What is the union of any set A with the empty set? What is their intersection?