Containment, proper containment, and why the empty set is inside everything

0:00Here is a school. Every dot is a student. And here, drawn inside it, is one class. Every student in that class is a student in the school. That is the whole relation, and you knew it before anyone wrote a symbol for it. Notice what it is not. It is not a claim about size. It is a promise about every single student in the class. Pick any one of them, and you will find that student in the school as well.

0:31Twelve students in the school. Five of them in the class. Five promises, and every one of them kept. Now try it the other way. Is every student in the school in that class? Walk along and find out. Student one, yes. Student two, yes. Student six, no. And you can stop there. Write that promise down and you have the definition. A set sits inside another when being in the first forces being in the second.

1:03One implication. Not a comparison of two sets, but a single statement about an arbitrary object. If it is in here, it is in there. Everything strange about this topic comes from reading it any other way. So here is the procedure the definition hands you. Take the first set. Walk its members, one at a time. For each one, ask the second set a single question. Is this a member of yours?

1:35If every answer is yes, the implication held in every case there was, and the containment stands. If any answer is no, you have stopped, and you are holding the reason. Watch it settle something. Fifty-six. Its divisors are one, two, four, seven, eight, fourteen, twenty-eight and fifty-six. Eight of them. Two of those eight are prime. Two, and seven. Do the primes sit inside the divisors? Two questions. Is two a divisor? Yes. Is seven a divisor? Yes.

2:13Done, and the containment holds. Now the other way. Do the divisors sit inside the primes? First question. Is one prime? No. One question, and it is over. That asymmetry is the point. To show a containment you must ask about every member. To show it fails you need exactly one, and the moment you have it you can stop. Containment can hold in one direction, or the other, or both, or neither.

2:45All four happen. Here are the vowels, and here is the set holding a, b, c and d. Do the vowels sit inside it? E is a vowel, and e is not there. No. Does it sit inside the vowels? B is there, and b is not a vowel. No. Neither direction, and one member each way is the entire proof. Now a different pair. The list one, three, five, and the odd numbers below six.

3:15Every member of the list is an odd number below six. Every odd number below six is on the list. Containment both ways. And that is not two facts about two sets. It is the statement that they are the same set, said twice. Push the definition somewhere it feels wrong. Does a set sit inside itself? Run the procedure. Walk its members, and for each one ask whether it is a member.

3:46Yes, obviously. You just took it out of there. Every answer is yes, so the containment holds. Every set sits inside itself, and there is no trick in it. This is why sitting inside does not mean being smaller. Smaller is a different word, and it is coming. What sitting inside means, and all it means, is that nothing in the first is missing from the second. Now push it the other way, at the set with nothing in it.

4:18Does the empty set sit inside the set one, two, three? Run the procedure. Walk the members of the empty set. There are none. The walk is over before it starts. The check finishes having asked zero questions. Not one question that got a lucky answer. Zero. And it never found a member that failed, because there was no member to find. So the containment holds. Not by agreement, and not by convention.

4:49There is no object available to break the promise, and a promise nothing can break is kept. The empty set sits inside every set there is. It sits inside itself. And run it once more in reverse. Which sets sit inside the empty set? Only the empty set. Anything with a member has a member that is missing. One, two, three sits inside one, two, three, four. And the other way round fails.

5:20Four is over there, and four is not here. So this containment runs in one direction only, strictly. There is a word for that. The first is properly inside the second, and the second is called the superset. Proper is the word that adds smaller. And now the empty set again. It is properly inside every set that has anything in it at all. And it is not properly inside itself, because it is itself.

5:52Careful with a habit you have from numbers. Any two numbers can be compared. One of them is at least as big. Sets are not like that. Take the set one, three. Take the set one, five, nine. Both of them sit inside one, three, five, seven, nine. Now put those two against each other. Does one, three sit inside one, five, nine? Three is missing. Does one, five, nine sit inside one, three?

6:22Five is missing. Neither. Two sets can both live inside a third and stand in no relation whatever to each other. Containment arranges some things and leaves others side by side. Here is the letter a. And here is the set whose only member is a. A box with one thing in it. They are not the same object, and everything that follows depends on seeing that. Is a a member of the set a, b, c?

6:55Yes. Is the box holding a a member of the set a, b, c? Look at what that set holds. It holds a, and b, and c. Three letters. It does not hold any boxes. So no. But does the box sit inside a, b, c? Walk its members. It has one. A. Is a in there? Yes. So yes, it sits inside. The same two objects, two different questions, two different answers.

7:26Now the case worth slowing down for. Three sets. The first holds the number one, and nothing else. The second holds two things. The first set, and the number two. The third holds three things. The first set, the number two, and the number three. Take a breath and read what is in each one. The whole difficulty is that one of these boxes contains a box. Is the first set a member of the second?

7:56The second's members are this box, and two. Yes. It is right there. Does the second sit inside the third? Walk it. Is the box a member of the third? Yes. Is two a member of the third? Yes. Two questions, both yes. So it does. So the first is a member of the second, and the second sits inside the third. What follows? Does the first sit inside the third?

8:26Walk it. Its only member is the number one. Is the number one a member of the third? The third holds a box, and two, and three. The number one is not among them. It is inside the box, which is not the same as being in the set. So no. And now the thing that usually gets left out. Something does follow, and it always follows. The first set is a member of the third.

8:55Why? Because the second sitting inside the third is exactly the promise that every member of the second is a member of the third. And the check, on its way past, already asked about the first set and got yes. Membership travels through containment. Containment does not. Those are two different statements, and only one of them is safe. Here is one set with four members. One, two, the set holding three and four, and five.

9:28Four things, and the third of them is a box. Eleven statements about it. Five are right. The box is a member. The set whose only member is that box sits inside. One is a member. The set one, two, five sits inside. And the empty set sits inside. Six are wrong. The set three, four sits inside. One sits inside. The set one, two, five is a member. The set one, two, three sits inside.

10:02The empty set is a member. And the set holding the empty set sits inside. Do not file those six together. They are wrong in three different ways. A student who cannot tell them apart will keep making whichever one they made before. Three of them put something on the left whose members are not all on the right. The set three, four does not sit inside, because three by itself is not a member.

10:31The box is. Two of them name an object that is simply not among the four members listed. And one of them is not a statement about two sets at all. One sits inside is a containment with a number on its left. A number has no members to walk. That one is not false in the way the others are. There is nothing there to check. Last question, and it changes how you write.

11:01List every subset of a set. Not the ones you notice. Every one. Start from nothing, and take the members one at a time. Before any member is considered, there is one subset. The empty one. Now consider the first member. Every subset you have becomes two. One without it, one with it. One becomes two. Consider the second member. Two become four. The third. Four become eight. Each member is an independent decision, in or out, and each decision doubles the answer.

11:38So a set with three members has eight subsets. And you know that before you write a single one down. A list that gives you seven is wrong before you read it. So. One, two, three. Eight subsets, and here they are. The empty set. Then the three with one member each. Then the three with two. Then the whole set. One, three, three, one. Eight. Notice the two that people leave out.

12:08The empty set is a subset. And the whole set is a subset of itself. Drop either one and you have seven, and seven is not a number the doubling ever passes through. And the smallest case of all. The empty set has exactly one subset. Itself. One, not none. One more, to put it together. Four sets. The solutions of x squared minus eight x plus twelve equals zero.

12:39The set two, four, six. The even numbers. And the set holding six alone. Factor the quadratic. X minus two, times x minus six. So the first set is two and six. Now arrange them. Six alone sits inside two and six, properly, because two is missing from the smaller one. Two and six sits inside two, four, six. Four is missing. And two, four, six sits inside the even numbers, which run on past.

13:13Four sets in a line, each properly inside the next. And containment carries you from the first all the way to the last. So. Containment is one implication about an arbitrary member, and everything else was that read carefully. To show it holds, ask about every member. To show it fails, produce one. It runs in both directions exactly when the two sets are the same set. It runs from every set to itself.

13:47And from the empty set to everything, because a promise with no case to test is a promise nothing can break. And here is the thing to carry. Belonging puts an object in a set. Containment puts a set in a set. They are not two words for one idea. Membership travels through containment. Containment does not travel through membership. When a set turns up inside a set, that difference is the whole question.