Philosophically and educationally there are a number of ways to introduce and teach mathematical ideas. Most mathematicians, however, prefer to use Sets as the basic structure in math, and build all ideas from there, using sets to define everything.

At first, mathematicians used to look at sets as just collections of things, but Bertrand Russell, the famous British philosopher and logician, showed that viewing sets as collections is a bit naïve—if a set gets “too big” it is possible that contradictions will result. Before we discuss what the problem is, we will use this simple or “naïve” view of sets and consider a set just to be a collection of things.

**SETS**

Here are some important ideas about sets:

1. In describing sets we often use the word “all” or “only.” Even if we use only one of those two words, or neither of them, we ALWAYS mean both. For example, if we talk about the set of *all* positive even numbers less than 10 we mean the set {2, 4, 6, 8}. (Notice, we generally put the members or elements of a set between two curly-cue brackets, and we put commas between the different members.) This set, {2, 4, 6, 8}, is both ALL the positive even numbers less than 10 and ONLY the even numbers less than 10. If we only say “the set of positive even numbers less than 10” and omit the word “all” we still mean this set. If we say “the set of *only* positive even numbers less than 10,” we still mean this set.

2. Repeating an item in a set, or changing the order does not change the set. For example, the sets {2, 2, 4, 4, 4, 6, 8} and {8, 4, 6, 2} and {2, 8, 4, 6, 8} are all equal to each other and the set mentioned in the previous paragraph.

3. There does not have to be any connection between items in the set or uniformity. For example, we can form a set that consists of all red Cadillac’s, Mexican immigrants, and stars that are less than 100 light years from the sun.

4. The things inside a set are called members or elements of the set. And there is a special symbol to so indicate: For example, we can write 2∈{2, 4, 6, 8} to mean that “two is a member of the set of all positive numbers less than 10.” We can also indicate something is *not* a member of the set by using the same symbol with a line through it. For example, 3∉ {2, 4, 6, 8}.

5. We can also talk about one set being inside another set—that is to say, if every member of one set is inside another set, we use this symbol:⊂ . For example,

{2, 8}⊂ {2, 4, 6, 8}. This may be read as “The set {2, 8}is contained in (or inside of) the set {2, 4, 6, 8}.” The set {2, 8} is called a subset of the set {2, 4, 6, 8}. Therefore, we can also read the above expression as “The set {2, 8}is *a subset of* the set {2, 4, 6, 8}.”

6. There are two types of subsets—proper and improper. If someone starts with a set and “removes” some elements and “keeps” some elements, then the result is a proper subset. For example, if the starting set is {2, 4, 6, 8} and one “keeps” 2 and 8 and “removes” 4 and 6, then what we have left is a *proper* subset, namely, {2, 8}. It is possible, though, to “keep” everything and “remove” nothing. In this case the “subset” is the full set,

{2, 4, 6, 8}. This full set, when thought of as a subset is called an *improper subset*. Similarly, one can “keep” nothing and “remove” everything, in which case the subset, is empty or has nothing in it. It is written as {} (empty brackets) or as . It is also called an *improper subset*, and called “the empty set.” If we want set A to be a subset of set B, and we allow that A could be all of B or the empty set—that is A could be a proper or improper subset–we write A B. If we do not want to allow the possibility that A could be all of B (but we still allow the possibility that A could be the empty set) we write

A B. If we want A to be a subset of B but we do not want to allow the possibility that A could be empty, there is no special symbol for that—we just add a sentence that “A is not empty.”

7. There is a difference whether we write 2∈{2, 4, 6, 8} or {2}⊂{2, 4, 6, 8}, even though they basically mean the same thing. 2 without any set brackets is just the number 2, and it is an element of the set. {2} is an entire set. You can look at is as being the number 2 put inside a basket or fancy package. This mathematical statement “{2}⊂{2, 4, 6, 8}” says everything in the “basket” on the left is also inside the basket on the right—even though the basket on the left has only one item. However, this statement “2∈{2, 4, 6, 8}” is not talking about two baskets. It talks about one basket only. It merely says the number 2 is a member of, or an item in, the “basket” on the right.

8. A set can contain a set, but if we are not careful, we can get into trouble (Russell’s Paradox). Here is an example of how a set can contain a set, without getting into trouble.

If we call S = {2, 4, 6, 8}, we can create a set T = {S, 1, 2, 3, 4}. How many members are there in T? There are 5. One of them is a “basket” which contains 4 items—namely the set S. But T only has 5 items in it, *not* 9, and not 7. Note that we can write

{2, 4, 6, 8} ∈ T just like we can write {1}∈ T or {2}∈ T, because the set S is a *member*, *not* a subset. Note that because 2 is in *both* S and T we can write 2∈ S *and* we can *also* write 2∈T (because 2 is also a member of T—a fact totally independent of whether or not S is also a member of T). Similarly with 4 which is both a member of S and of T. However, with 6, we can write 6 ∈ S, but we* cannot also* write 6∈T, because 6 is *not* a member of T. Now we can also write 2∈ S ∈ T, which means “the element, 2, is a member of set S, and the set S is a member of the set T.” Note that this last statement does not say whether or not 2 is in T. It merely says that 2 is in S, and that S is in T. Similarly, we can write 6 ∈ S ∈ T, which means that 6 is in S and that S is in T. It does **not** say whether 6 is in T or not. **Be sure to spend the time necessary to understand ***carefully and exactly*** each of the examples just mentioned in this paragraph before going on. Do not skim, and do not just read it. Understand it slowly and carefully in detail.**

9. Now we will explain how the idea of a “set containing a set” can be problematic.

To start, first notice that many sets, even if they are infinite in size, can generally be expressed in English sentences containing a few words. For example, “The set of all even numbers” is a sentence describing an infinite set—the even numbers—and the description should be very clear to anyone, even someone in the second grade. Yet this description only takes six words! (Side point: In general mathematicians like to be very terse, and the brevity of mathematical statements can confuse people who are not used to paying attention to the detail and exact meaning of every word—an absolute necessity in mathematics.) In spite of the tendency towards terseness, there are some sets that might take many words to describe, for example, consider this set: “Let K = the set of all people who (1) were born in New York, (2) have red hair, (3) moved to California after age 10, (4) drive a Ford, (5) did not graduate college, (6) like to follow football games on television, (7) like Big Macs, (8) own a computer, etc. “ If I were to list 100 different conditions, even using terse wording, the description would certainly be over 100 words long. Of course, it could be there is no person on earth who meets all 100 conditions, especially if some of them are rare, such as (over 6 feet 5 finches tall). While that one condition might be met by some people, the number of people meeting this condition might be relatively small, so it could be there is no one who meets all 100 conditions. If this is the case, then K, the set described is the empty set, but we would not know that without a survey unless we created a ridiculous condition, such as the person must be over 10 feet tall. In short, this set could very well not be the empty set, but the description could be very long—over 100 words.

Now let H be the set of all sets that can be described in an English sentence containing less than 100 words. By definition, the elements of H are themselves sets. Clearly, the set K described above is not inside of H, because its description is too long. However, all other sets mentioned above are sets that are inside of H—as well as many other sets, such as the set of real numbers, the set of rational numbers, the set of all people on earth, etc. These sets are infinite, but their descriptions are short, so they are all inside of H. In fact, the set H is also inside of H because the description of H above contained 17 words (which is certainly less than 100). Thus we have something like the following H ={H, N, Z, Q, S, T, ….} H is in fact a member of H! This concept is counter-intuitive, because if we think a set as a collection, a collection cannot obviously contain itself. So while it is nice to think of a set as a collection, the analogy breaks down when we deal with more complicated examples, as is so in this case.

Now, here is Russell’s Paradox. All the sets we have above, or “standard” sets, such as whole numbers, natural numbers, integers, etc. we will call “ordinary” sets. However, any set, such as H, which contains itself, we will call an “extra-ordinary” set.

Let Obe the set of all ordinary sets. Question: Is O ordinary or extra-ordinary? We will find we have a problem (contradiction) or paradox, either way we look at it. If O is ordinary, then since O contains *all* ordinary sets (by its very definition) then it must contain itself (since it is ordinary)—and by containing itself it would then become extra-ordinary—a contradiction! Again, if O is extra-ordinary, then O must contain itself, by the definition of the word “extra-ordinary.” But this fact contradicts the definition of O , since O is supposed to contain *only* ordinary sets, not an extra-ordinary ones! So as a result, we cannot decide whether O is ordinary or extra-ordinary.

The solution to this paradox, is to redefine or rethink what we mean by sets. They are not simply any collections, but they must be limited so they do not grow too big. There are several ways to achieve this re-definition—but most are too sophisticated to explain here. We will explain, briefly (not in detail) the simplest method here.

We start by defining some set, which we call U, the universal set. U could be the real numbers, natural numbers, or any other set you want it to be. However, we cannot just create “new” sets at random. They must be formed by certain rules—for example, by taking elements from U, by taking subsets from U, or by taking unions or intersections, or complements of these sets. So while we can make an infinite number of sets, and we can even create a set of subsets, we can’t create “higher levels” of sets containing sets, so we stay clear of this paradox.