Difference and complement: the same idea with and without a universal set

0:00Here are two sets, and here is a third built from them. It holds the members of the first that the second does not have. That is the difference, and the whole of it is two conditions asked of one object at a time. Are you in this one. And are you outside that one. Answer yes to both and you are in the difference. Answer no to either and you are not.

0:26Notice at once that the two conditions are not the same shape. One set has to be entered and the other has to be avoided. Hold on to that, because almost everything in this video comes out of it. Take the set holding one, two, three, four, five and six. And the set holding two, four, six and eight. Walk the first one and ask the two questions of each member.

0:54One is in the first and outside the second, so it stays. Two is in the first, but it is also in the second, so it goes. Three stays. Four goes. Five stays. Six goes. What is left is one, three and five. Now turn the question round and take the first away from the second. Two, four and six are all in the first set, so all three go.

1:20Eight is not, so eight stays, and eight is the whole answer. Three members one way round and one member the other. Two different sets, from the same two sets, in the same operation. Union does not care which set you name first. Neither does intersection. So it is a reasonable guess that no set operation cares, and the guess is wrong. Look again at the two conditions. In the first set, and outside the second.

1:52Swap the two sets over and you have swapped which one is entered and which one is avoided. The condition is not symmetric, so the answer has no reason to be. Two examples will not settle a claim about every pair, so this one was settled by asking every pair. In a universe of five objects there are thirty two subsets, and one thousand and twenty four ordered pairs of them.

2:20On nine hundred and ninety two of those pairs, taking one away from the other gives a different answer each way round. The two answers agree on exactly thirty two pairs, and those are exactly the pairs where the two sets are equal. On every one of those, both differences are empty, so agreeing is all they can do. Order matters for difference in every case where there is anything to disagree about.

2:51There is a phrase people use that quietly hides all of this. They say taking B away removes the elements of B. It does not. It removes the members of B that were in A, and it has nothing to say about the rest of B. The eight in that second set was never in the first set at all. Nothing was done to it, nothing was removed, and it is sitting in the other answer untouched.

3:20The operation does not reach into the second set and delete things. It walks the first set and lets members through, or does not. Say it that way and the asymmetry is obvious, because only one of the two sets is ever walked. Now something the two operations do together. Take any two sets and form three things. What the first has and the second does not. What they share.

3:49And what the second has and the first does not. Those three never overlap. Not usually, not for nice sets — never, and here is the reason rather than the observation. Anything in the first piece fails the test for the second set. Both of the other two pieces require membership of the second set. So a member of the first piece is barred from both, and the same argument runs for each pair of pieces in turn.

4:20Together they are the union, with nothing left over and nothing counted twice. Which means the three sizes add up to the size of the union, on every pair of sets there is. It is worth being careful with the word three there. The three pieces always exist and never overlap. But they do not always have anything in them. Take two sets that share nothing and the middle piece is empty.

4:50Take one set sitting inside another and the first piece is empty. Of those one thousand and twenty four pairs, only three hundred and ninety have all three pieces occupied. That number is not a curiosity, and it can be got a second way. Every object independently lands in one of four regions, which is where the one thousand and twenty four came from. Strike out the arrangements that leave a named piece empty and three hundred and ninety is what survives.

5:24So on more than half of all pairs, the picture you should have in your head is not the standard one. Now fix a set that everything in the discussion lives inside, and call it the universe. Take some set inside it, and take that set away from the universe. What is left is called the complement. That is the entire definition, and it is worth being blunt about what it is not.

5:53It is not a new operation. It is the difference you already have, with the universe nailed to the left-hand side. Drawn, it is the rectangle with a hole in it. Every question you can ask about a complement is a question about a difference where the first set has stopped moving. And that one change, from a set that varies to a set that does not, is what the rest of this video is about.

6:22Here is a case that looks impossible until you see what it is doing. Let the universe be the prime numbers, all of them, going on for ever. And let the set be the primes that do not divide forty two. Forty two is two times three times seven, so the primes that divide it are two, three and seven. Every other prime is in the set. So the set is endless, and what lies outside it has three members.

6:51Widen the range you are looking at and the set gets bigger, while the outside stays at exactly three. An endless set does not have to have an endless complement, and nothing here is strange. The outside is small because the universe was already narrow — everything in it was a prime before we started. That last point deserves its own scene, because the universe is usually left unwritten. Take the whole numbers from one to ten, and the odd ones among them.

7:26Outside the odd ones are two, four, six, eight and ten. Five members. Now keep the same set of odd numbers and widen the universe to twenty. The set has not changed by a single member. But the outside is now fifteen members, and it is a different set than before. Or take a class of six students and the three girls in it. Outside them are the three boys.

7:53Add two more students to the room and the answer changes again, without anyone touching the set of girls. A complement is never a property of one set on its own. It is a fact about a set and a universe, and only one of those two usually gets written down. Complement comes with four rules, and they are usually presented as things complement can do that difference cannot. A set together with what is left of it gives the universe back.

8:26A set met with what is left of it is empty. Taking it away twice gives the set back. And the two extremes swap over. Now write those four as statements about difference, with the left-hand set free to be anything, and put them to all one thousand and twenty four pairs. Two of them hold on every single pair, with no condition attached at all. Being met with what is left of it is empty, always.

8:56And the two extremes behave, always. The other two fail, on seven hundred and eighty one pairs each. Not seven hundred and eighty one each in different places — on exactly the same pairs, both of them. And every one of those failures is a pair where the second set is not inside the first. Turn it round: on the two hundred and forty three pairs where it is inside, both laws hold, every time.

9:26Two hundred and forty three and seven hundred and eighty one is the whole thousand and twenty four, so containment is not a sufficient condition, it is the exact one. So the complement laws were never gifts of complement. They are difference laws that need the second set to be inside the first. And the universe is the one left-hand set that satisfies that for everything in the discussion, by definition of the word.

9:59Watch the failure directly. Take a set away from another, and then take the result away from that same set again. You do not get back what you removed. You get what the two of them shared. And what they shared is what you removed exactly when what you removed was inside the set to begin with. With the universe on the left that condition is free, so complementing twice returns the set, every time.

10:28That is not a deeper fact about complement. It is the same fact about difference, standing on ground where the hypothesis costs nothing. Complements do not need a list of members. A condition is enough, and the complement is the condition denied. Over the whole numbers, outside the even ones are the odd ones, and the other way round. Outside the multiples of three are the numbers three does not divide.

10:59Outside the numbers divisible by three and by five are the numbers fifteen does not divide. Outside the perfect squares are the numbers that are not perfect squares, and likewise for cubes. A condition can also pin down a single number. The numbers where x plus five is eight — that is just three, so its outside is every whole number except three. Where two x plus five is nine, the set is just two, and everything else is outside it.

11:32The numbers at least seven leave one to six outside. And two x plus one greater than ten forces x above four and a half, so the smallest that works is five, and one, two, three and four are outside it. There is one of those that almost everybody gets wrong, so it gets its own moment. What lies outside the prime numbers? The answer people give is the composite numbers, and it is short by exactly one member.

12:04One is a whole number. One is not prime. So one is outside the primes, and one is not composite either. Outside the primes are the composites together with one. This is not pedantry about a special case. It is what the definition of complement says: outside means everything in the universe the set does not hold, and the universe here was the whole numbers. Whenever you complement a condition, ask what the universe was, and then check the awkward ones at the edge.

12:41So here is the shape of it. Difference asks two questions that are not the same question, which is why swapping the sets changes the answer. Two sets always split into three pieces that cannot overlap and together make the union, though on most pairs at least one of those pieces is empty. Complement is that same difference with the universe on the left, so it is not a second thing to learn.

13:11The rules that look like they belong to complement are difference rules that need one set inside the other. Two of the four never needed it. The other two need it every time, and the universe hands it over for free. A complement is a fact about two things, and one of them is usually not written down. Change it and the answer changes, without the set moving at all.

13:38None of that had to be memorised. All of it came out of asking, of one object at a time, whether it is in this one and outside that one.