Lesson 2 Solutions

I figure that if you’re going to take the time to do and/or think about the exercises, I might as well provide some solutions so that you can have the confidence of knowing that you’re doing it right!

Exercise 1)   Try to define some crazy sets, and try to determine if two sets are the same or not.  For example, can we define the set of all ideas?  What about the set of all thoughts?  What makes two different thoughts distinct? (Because remember, elements in a set need to be distinct!)  Is the set of all ideas equal to the set of all thoughts?  It depends on how you define them!  Don’t get too worked up about this stuff yet though—I’m just trying to show you that “mathematical thought” is indeed much broader than you could have ever imagined!

Solution: Well, this exercise doesn’t really have an exact “solution”, since you’re really just supposed to think of some examples yourself, but I’ll get the ball rolling.  Some crazy sets that one could define are the following: the set of all toenails (crazy just because it’s gross), the set of all universes (each universe is a single element—is there more than one element? are there infinitely many elements?), and the set of all sets!  The set of all sets is simply the set whose elements are themselves sets, and moreover it contains all of the sets.  We’ll discuss this wacky set a whole lot more here and in lesson 5.

Now, the part of this exercise asking about the set of ideas vs. the set of thoughts is really there to show the importance of having rigor in one’s definitions.  We need to define what we mean by “a thought” and “an idea” before we can ask whether or not these two sets are the same.  Let me make the following definitions (although it should be clear that there is no right or wrong definition, nor any exact, mathematical definition).  A thought is any reaction, feeling, emotion, opinion, observation, or any other mental process, either verbalized or not, by any sentient being anywhere.  Now, this is of course a complex issue because now we’d have to define every term in that definition—namely “feeling”, “emotion”, “sentience”, etc.—but let’s just go with this.  An idea, on the other hand, is any mental facility brought forth by any sentient being from the motivation to accomplish any kind of goal.

With these (highly inexact) definitions, it is clear that the set of all ideas and the set of all thoughts are not the same, because there are several thoughts that are not ideas.  It is the case, however, that any idea is indeed a thought, simply because an idea is a mental facility (of a particular kind).  For example, I just thought “I am hungry”, which is therefore a thought.  That thought is not an idea, however, because there is absolutely no goal that motivated its existence.  Now I have thought “I should eat”, which is an idea, because it was motivated by my wanting to accomplish the goal of no longer being hungry.

Clearly there is a lot of wiggle room here, and this is open to much (welcomed) debate and scrutiny.  The point that I’m trying to show, however, is that we sometimes need to work really hard to make definitions that are precise enough for us to actually be able to work with. Most mathematical definitions are easier than these two!

Exercise 2)  Can you think of a set that contains itself as an element?  (If you can’t, don’t worry, this one is tricky and it’s something we’ll be addressing more in a little while).

Solution:  If you haven’t already guessed it, an answer is actually in the solution to exercise 1 above.  After a little thought, we notice that the set of all sets contains itself!  It is, after all, a set, and since it’s the set of all sets, it must have itself as an element.  On the face of it there’s nothing logically wrong with a set containing itself, but as we explore sets of sets more deeply in lesson 4 and lesson 5 we’ll find that there is indeed something very wrong with this fact.  Nonetheless, it works to solve this problem and so we’ll go with it.  If this doesn’t make all the sense in the world at the moment, don’t worry, because we’ll be addressing this issue more slowly in the coming lessons.

Back to Lesson 2

On to Lesson 3


14 Responses to Lesson 2 Solutions

  1. Filip says:

    In solution to exercise #2 this set of all sets seams impossible. Since it is a set of all sets it has to contain itself but at this moment it also needs to contani sets of all sets with itself within it. It seams like a type of infinite loop situation. Is my reasoning correct?

    I also would like to sincerely thank you for making this site. Being CS major mathematics curriculum is design really poorly and extremely limited. Your website makes it possible for me (and many others) to see what maths really is.

    • Hi Filip,

      You’re definitely right to take issue with this “solution”, because as you’ll see in later lessons (as mentioned in this solution) we’ll find that these kinds of sets can’t really exist. The whole problem is indeed closely related to what you said, which is the logical conundrum we find ourselves in when sets can contain themselves as elements. As for the infinite loop, I’m not so sure I see what you’re saying. After all, it’s the set of all sets, so it will also contain all sets that contain themselves as elements, as well as all sets that contain the set of all sets. But still, as it stands, there’s nothing really wrong with this. It’s obviously a wildly infinite set, and it is contained in many of the sets that form its elements, but it’s “the set of all sets”, so it’s bound to be pretty gigantic and weird. Of course, this is tough to really give a solid answer to, because the whole point of “the set of all sets” is to be ill-defined. That said, I don’t think the illness of its definition comes from this type of infinite loop, but rather from other closely related issues (again, explored in lessons 4 and 5).

      And as for the second part of your comment: thanks! I’m glad you’re finding it helpful and interesting, and I do hope to be getting some more content up very soon. I hope you keep reading!


  2. Anonymous says:

    Very cool! Just found your blog today and dove right in. I’ve always suspected that I have been missing out on some kind of interesting/stimulating/profound ways of thinking by having only a rudimentary understanding of math. I get the feeling you’re about to show me just how much I’ve been missing – thanks!

  3. jpsk says:

    I chance upon your site whilst googling for subsets and vacuously true statements, as i have a hard time wrapping my head around the definition of subset, where A is a subset of B means that for all x, if x is an element in A, then x is an element in B, which also means that (x is not an element in A) OR (x is an element in B). The latter statement made my head spin a bit, as I’ve no idea how vacuously true statements could be really true, and why it had to be true rather than inconclusive.

    I’m beginning to feel excited upon reading your site, and decided to try the exercises which leaves me here, thinking about the sets that contains itself as an element, and my best thought would be a box of boxes of ice cream, which seems recursive and endless, since upon opening a box of boxes of ice cream, one will find more boxes of boxes of ice cream, and one never seems able to find the ice cream but the boxes. Just my thought, and i’ve not read the rest of the site yet, but i believe i will as i’m just excited to pen my thoughts. Once again, thank you.

    • I’m glad you’re liking the site, and I hope you continue to do so! As for your analogy with boxes of ice cream, the issue is a bit more subtle than that. Namely, a set containing itself is different than a box of boxes of ice cream, in the following way. A box full of boxes of ice cream is such that the box containing the other boxes is different than the boxes it contains. In other words, suppose we have a box, call it BOX1, and inside this box there are other boxes. Suppose BOX1 contains BOX2 and BOX3. Then BOX2 and BOX3 are different boxes than BOX1. Thus BOX1 contains two elements, and both of those elements are different from BOX1 itself. A set containing itself means that one of the elements IS the set itself. I.e., it would be the case that, for example, BOX1=BOX2. Now, intuitively this makes no sense when we think about physical boxes. However, when considering sets, we can hypothetically consider “the set of all sets”. Now, “the set of all sets” is itself a set, so it literally contains itself as an element—one of its elements is itself! These are the kinds of issues one needs to avoid when talking about sets, and this is the sort of issue we allude to in this and the following lessons. Read on, and let me know what you think!

  4. neyla says:

    Hi there ! This is actually very awesome! I had a very bad issue with math before, but this year I tried to be more open about it and I can see that it’s actually beautiful if you look more deeply. Sadly, the education system in my country is bad as well and nobody ever teaches you to love mathematics. I’m really glad I have discovered this website, it gives me hope!

  5. Yuanyuan Sun says:

    I think the solution to question 2 can also be: A={ All sets that contains 1 element }, or add other element to A like: all set that contains 2 elements and so on…

    • Ah, you’re very close! Unfortunately, the set of all sets that contain 1 element is not a set that contains itself as an element (and the same is true for the set of all sets that contain 2 elements, etc.). The reason for this is simply that there are for more than 1 set with 1 element. For example, {a}, {b}, {Donkey}, and {5} are all sets that contain one element. Therefore, if we denote “the set of all sets that contain one element” by, for example, ONE, then it turns out that ONE={{a}, {b}, {Donkey}, {5}, …}, where the “…” is there because there are MANY more sets that contain one element. Thus, we see that ONE is not a set that contains one element, but rather it has many, many, many (infinitely many) elements! Thus, ONE does not contain itself as an element.

      You are, however, dangerously close to the definition of a set that contains itself as an element. Namely, all you need to do is tweak a couple of words in the above definition. Can you see how? If not I’ll happily help out more 🙂

      And sorry for my late reply, I hope you keep enjoying the site!

  6. Using your definition about ideas, what about the set of all ideas and the set of all sets, giving that there exists a goal to be thinking about the set of all ideas; My words about the latter ones are the following:

    >> The set of all ideas is part of the set of all sets.
    >> The set of all sets is also part of all ideas.

    The set of all sets is indeed an idea for me, it is given to accomplish the goal of reunite all sets in one. Please recommend me something to fix the explanation below.

    Thanks for your resources.

    • Interesting! Based on your notion that “the set of all sets” is indeed an idea, then it is indeed the case that “the set of all sets” is an element of the set of all ideas. And clearly the set of all ideas is an element of the set of all sets (since the set of all sets contains EVERY set). There’s nothing immediately wrong with this, it’s just interesting! Of course, the set of all sets is ill-defined (as we see in lesson 5), and the set of all ideas isn’t defined either (because there is no way for us to REALLY and rigorously define an idea), but based on the “definitions” that we’ve gone with then indeed you’re right. Neat!

  7. taorkon says:

    What about an empty set? Empty set contains nothing > ES = {} , {ES} = {}? Or is it {ES} = {{}}? If it s the second case, what’s the difference? I guess it does differ in programming, but does it really matter in our situation?

    • Great question! This one is indeed subtle. The empty set {} does not contain itself as an element. The set {{}}, on the other hand, is a set with one element: the empty set. Namely, the set {{}} is a set of sets, and it has one element. It is like the set {{a}, {a,b}, {a,b, elephant}}, which is a set of sets. This set has three elements: the set {a}, the set {a, b}, and the set {a, b, elephant}. There’s nothing wrong with having sets of sets. Namely, the set {a,b} is one element in the aforementioned set, and it is itself a set with two elements. Similarly, the set {{}} is a set with one element, the empty set. This set that {{}} has as an element just happens to have zero elements. Does that make sense?

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