Why nothing can be complemented until the surrounding set is fixed

0:00Here is a question that sounds completely answerable. Which numbers are not even? Ask three people and you will get three answers, and every one of them will be correct. The first says the odd numbers. The second says the odd numbers, and every fraction as well. The third says the odd numbers, every fraction, and every irrational number too. Nobody has made a mistake. The question simply did not have one answer, because one word in it was never given a range.

0:35That word is not. Not even means everything that is not even. Everything. And the moment you try to collect everything, you have to answer a question nobody asked out loud. Everything in what? So we answer it first, before anything else happens. We name a set, and we agree that for the rest of this conversation nothing outside it exists. Inside the whole numbers, not even picks out the odd numbers.

1:08Inside the fractions, it picks out the odd numbers together with every fraction that is not whole. Same three words, different sets. The surrounding set is not something you can work out from the set you started with. It is handed in separately, and it has to be handed in first. That surrounding set has a name and a symbol. We call it the universal set, and we write it as a capital U.

1:37It also has a picture, and the picture is the clearest thing in this whole topic. Draw a rectangle. That rectangle is U, and everything in the discussion lives inside it. Now draw a circle inside the rectangle and call it A. The region left over, inside the box and outside the circle, is the complement of A. Everything that is not in A. And now look at what the picture shows that the words hide.

2:08The complement is a region trapped between two boundaries. Move the outer boundary, and that region changes shape, even though the circle never moved at all. Let us do exactly that, with a set you already know. Take A to be the integers. Put them inside the rational numbers, and ask what lies outside. The answer is every fraction that is not a whole number. One half is out there, and minus seven thirds is out there.

2:41Now put the very same integers inside the real numbers instead, and ask again. One half and minus seven thirds are still outside, but so is the square root of two, and so is pi. Nothing about the integers changed between those two questions. Not one member was added, not one was removed. The set outside them changed because the surrounding set changed, and for no other reason. Here is a case where the two sides come out lopsided in a way worth seeing.

3:17Let the universal set be the prime numbers, all of them. And let A be the primes that do not divide forty two. What lies outside A? Forty two is two times three times seven, and those three primes are the only ones that go into it. So the set outside A has exactly three members. Two, three and seven. Meanwhile A itself never runs out. Hand me any finite list of its members and I will build you one more, by multiplying your list together with those three, and adding one.

3:54The prime that falls out of that cannot be on your list, and cannot be one of the three either. An endless set, and a complement you can write on one line. None of this is about numbers. Take a class of eight students, and let A be the ones who play chess. Say there are two of them. Inside the class, the set outside A has six members. The other six students.

4:22Now widen the universal set to the whole school, twenty students, and ask the same question about the same two chess players. Now the set outside A has eighteen members. Six and eighteen, from one unchanged set A. In population studies the surrounding set is everyone alive, and that is a real choice somebody made, not a mathematical fact. A universal set is a decision about what the conversation is about.

4:56So if the universal set is a decision, is any decision as good as any other? No, and there is a test. A universal set has to be big enough to hold every set the discussion is going to mention. Every single one, completely, with nothing hanging outside. Call that adequacy. And notice how cheaply it can fail. To accept a candidate you must check every member of every set.

5:25To reject one you name a single member of a single set that it does not contain, and you are finished. One escaping member kills a universal set, no matter how much else it holds. Let us run that test properly. Here are three sets. A holds one, three and five. B holds two, four and six. C holds zero, two, four, six and eight. And here are four candidates for the universal set, offered one at a time.

5:59The whole numbers from zero to six. The empty set. The whole numbers from zero to ten. And the whole numbers from one to eight. Before we check, notice that three of those four look perfectly reasonable. Only one of them survives. Start with zero to six. A fits, B fits, and C is fine until we reach its last member. Eight is in C, and eight is not in the candidate.

6:29Refused, and we needed exactly one number to do it. The empty set holds nothing at all, so it fails on the first member we ask it about. Zero to ten holds all three sets completely, so it is accepted. And one to eight? This is the interesting failure. A fits and B fits, so it is already holding two of the three sets. But C contains zero, and this candidate starts at one.

6:59Zero escapes, and that is the end of it. It is very tempting to think a big enough set will always do. So let us kill that idea properly. Put the three sets together into one collection, and count it. Zero, one, two, three, four, five, six and eight. Eight members. Now count the candidate that just failed, the whole numbers from one to eight. Also eight members. Exactly the same size, and opposite verdicts.

7:33The first one works because it was built by collecting everything that needed holding. The second fails because it is missing zero and carrying a seven that nobody asked for. Size is not the property being tested. Containment is. That question had one correct answer out of the four on offer, and that is easy to misread. It does not mean there is one correct universal set. Stay inside the whole numbers from zero to ten and there are two thousand and forty eight sets you could choose from.

8:10Check every one of them against the test. Eight of them pass. Every one of those eight contains the collection we just built, and that collection is the smallest of them. The other seven are that same collection with some of seven, nine and ten added on. Those three numbers are simply free. Carrying them costs nothing, because nothing in the discussion ever asks about them. Adequate is the requirement.

8:40Smallest is not. Here is that freedom doing real work. Suppose one discussion is about right angled triangles, and another is about isosceles triangles. What should surround them? Take triangles with whole number angles, just so we can count. Fix the right angle and the other two share what is left, which gives forty five right angled triangles. Choose the angle that repeats and the odd one is whatever remains, which gives eighty nine isosceles ones.

9:18Two families, different sizes, and both sitting inside the triangles. So the triangles serve as a universal set for both discussions at once. And here is the part that is easy to miss. Those two families are not separate. The triangle with angles ninety, forty five and forty five is in both of them. The triangles are not the only thing that works here. All polygons would work. All plane figures would work.

9:52Each of those holds both families completely, and that is the whole requirement. A question like this has many right answers, and a student who produces a different one from yours has not made a mistake. But the freedom is not unlimited. Try to use only the right angled triangles as the surrounding set, and the test refuses immediately. The isosceles family contains triangles with no right angle in them at all, and every one of those escapes.

10:26Too small is a real failure. Larger than necessary is not. Now step back and look at what this changes. From here on, every time you take a complement, a universal set has already been fixed. Sometimes it is written down. Far more often it is sitting in the background, agreed to silently by everyone in the room. And that is fine, right up until two people are silently assuming different ones.

10:58Then they compute two different complements from the same set, both correctly, and disagree without ever finding out why. So when a complement appears without a stated universal set, that is the question to ask. Not what is outside A. Outside A, within what? There is an obvious way to avoid all of this. Why not fix one universal set once and for all, containing absolutely everything, and never think about it again?

11:32In nineteen oh two, Bertrand Russell showed that you cannot. Suppose there were a set of all sets. Then we can describe a particular one. Let R be the set of all the sets that are not members of themselves. Now ask one question about it. Is R a member of R? There are exactly two answers to try, so let us try both. Say yes, it is a member of itself, and the rule says it must be left out.

12:02Say no, it is not a member of itself, and the rule says it must go in. Both answers contradict the rule that defined it. The rule itself is perfectly harmless for anything else. What breaks is the assumption that there was a collection of all sets to pick R out of in the first place. So there is no everything, and that is why the universal set is chosen fresh each time.

12:31Not because mathematicians are being fussy, but because the alternative genuinely does not exist. Which brings the whole topic back to one habit. Before you ask what is outside a set, say what you are inside. The universal set is not a detail of the answer. It is one of the inputs to the question.