Turning a claim about sets into a picture you can read off

0:00Somewhere in every course on sets, someone draws two overlapping circles. And almost everyone treats that drawing as a helpful sketch, with the real work happening somewhere else. This video is going to argue the opposite. Drawn correctly, that picture is not an illustration of an argument. It is the argument. But correctly is doing a lot of work in that sentence, and by the end you will be able to look at a diagram and say whether it proves anything at all.

0:35Start with the parts. A rectangle stands for the surrounding set, the one everything in the conversation belongs to. Inside it, each closed curve stands for one subset. A dot inside a curve is a member of that set. And the rectangle is not decoration. Without it there is nowhere to put the objects that are in none of the sets, and that region is exactly where every complement lives.

1:05Three parts, and each one carries a job. Here is the smallest useful example. Let the rectangle hold the whole numbers from one to ten, and let one curve hold the even ones. Two, four, six, eight and ten go inside the curve. One, three, five, seven and nine go in the rectangle, outside it. Five inside, five outside, and every one of the ten appears exactly once. That completeness is not a nicety.

1:38If a number had nowhere to go, or two places it could go, the picture would not be answering the membership question. And notice what the drawing does not tell you. The curve could be drawn large or small and nothing would change. Area carries no information at all. Now add a second curve, and put it entirely inside the first. Let it hold just four and six. Four and six sit in the inner curve.

2:09Two, eight and ten sit in the ring between the two curves. One, three, five, seven and nine sit outside both. Three places, and again all ten numbers accounted for. Now look at what you can read off without anyone writing a symbol. Nothing in the inner curve is outside the outer one. That is containment, and you did not have to be told. You also get the other half for free.

2:37The outer curve is not inside the inner one, because two is in the ring, so the two sets are not the same. So why does any of this work? Here is the whole idea, and it is short. Take any object in the rectangle and ask it one question per curve. Are you in this one? Yes or no. With two curves that is two questions, so there are four possible answer patterns.

3:06In both, in the first only, in the second only, in neither. Every object in the rectangle has exactly one of those four patterns, and there is nothing else it could have. Now here is the requirement. If the curves are drawn so that every pattern has its own region, then the regions are the cases. Not examples of the cases. All of them. Let us count. Two curves drawn overlapping cut the rectangle into four regions, and we just said there are four patterns.

3:44One region each. Add a third curve crossing both of the others. Now every object faces three questions, so there are eight patterns, and the drawing gives eight regions. Still one each. You can check that on the picture by walking round it and naming the pattern each region stands for. The one in the middle is in all three. The corner of the rectangle outside every curve is in none of them.

4:14And the six in between are the mixtures. Now the payoff. Suppose you tint the region an expression describes. What have you actually done? You have answered that expression's membership question on every possible case, simultaneously, because the regions are the cases. So take two expressions and tint each one. If the tinted areas match, the two expressions agree on every pattern there is, and they are the same set.

4:47If the areas differ, there is a region where they disagree, and that region is a case where one holds and the other does not. That is not a hint that they differ. It is a counterexample, and you can point at it. Which brings us to the failure, and it is worth being precise about it. Draw two circles that do not touch. That is the correct picture of two sets with nothing in common, and there is nothing wrong with it as a picture.

5:20But count its regions. Inside the first only, inside the second only, and the rest of the rectangle, in neither. Three regions for four patterns. One pattern has nowhere to live, and it is the overlap. You will often hear that two regions are missing here. That is not right, and the difference matters, because which region is gone is exactly what the drawing then gets wrong. So what does one missing region actually cost you?

5:54With two curves there are sixteen different expressions you could write, counting every way of answering yes or no to the four patterns. On the drawing with the overlap missing, eight pairs of genuinely different expressions become impossible to tell apart. Here is one of them. The union of the two sets, everything in either one. And everything in exactly one of them, but not both. Those are different sets, and the thing that separates them is an object in both.

6:29Take the overlap away and they tint identically. The picture would let you prove they are equal, and they are not. Now do the other thing people do, and rub out the rectangle. Two bare circles, floating. A second pattern loses its home, and it is the one for objects in neither set. So now two regions really are missing, and the second one is the region every complement lives in.

6:59Which means a drawing without a rectangle cannot check a single claim that mentions what is outside a set. The rectangle looked like a frame around the picture. It was carrying a case. Let us watch the method do real work. Three curves, all overlapping, and a law to check. Take everything in the second or the third set, and tint it. Now tint the part of the first set that meets it.

7:29Hold on to that shape. Start again with fresh panels. Tint the overlap of the first set with the second. Then the overlap of the first with the third. Then both of those together. And that final shape is the one we were holding on to. The two shadings are identical, and that identity is the distributive law. You can read more off those panels than the verdict. The region the law is about is three of the eight, so the law is not a statement about everything.

8:05It says something on three cases and nothing on the other five. The two overlap panels cover two regions each. And they share exactly one, the piece in all three sets. Two plus two, sharing one, gives three. That is why the answer is three regions and not four, and it is the reason the law is stated with a union on the outside. The picture is not just confirming the law.

8:33It is showing you the arithmetic the law is doing. The same method settles two more claims in about a minute. Four expressions, two curves. Outside the union of the two sets. Outside the first and outside the second at the same time. Outside the overlap. Outside the first or outside the second. Shade them and the answer is immediate. The first two tint the same single region, the one in neither curve.

9:06The last two tint the same three regions, everything except the middle. And the two pairs are not the same as each other, because one region against three is not a close call. One more thing, and it explains something you may have noticed. The curves are always described as usually circles, never as necessarily circles. There is a hard reason for that. Two circles can cross at most twice.

9:36So when you add a new circle to a drawing, the ones already there can cut it into only so many arcs, and each arc splits one region in two. Work that through and n circles can produce at most n squared, minus n, plus two regions. One circle gives two regions, two give four, three give eight. Those are exactly the numbers we needed. But four sets need sixteen regions, and four circles can only manage fourteen.

10:08Circles are not a convention that happens to be popular. They are provably enough for three sets and provably not enough for four, and past that the gap only widens. Finally, the limits, because a method you cannot see the edge of is a method you will misuse. A curve can stand for an endless set, but no drawing lists its members. The dots are a device for small finite examples and nothing more.

10:40You cannot count members off a diagram unless somebody wrote them in. And here is the one that catches people. A set can be a member of another set. Take the collection holding the number one and holding the pair two and three. That collection has two members, not three. Two and three are not members of it. A dot cannot show that, because what belongs in that place is a curve, not a dot.

11:10So here is where we have arrived. A diagram drawn so that every combination has a region is a complete case analysis that you can look at. Tinting is checking, and matching tints is a proof. A diagram drawn to a special arrangement is a picture of one situation, and it will happily agree with things that are false. The difference between the two is not artistic. It is whether every pattern got a region.

11:42So before you trust a diagram, count its regions and count the patterns. If those two numbers match, you can read the answer straight off the page.