"Or", "and" and "not" are the three set operations under new names

0:00An event is a subset of the list of outcomes. Sets already combine. So when you want to say NOT this, or THIS OR THAT, or THIS AND THAT, nothing new has to be invented: the operations are already sitting there, waiting. That is the whole of what follows, and it is usually presented as a table to memorise. It is not a table. It is a claim, and a claim can be wrong.

0:29The claim is that combining two DESCRIPTIONS with an ordinary English word and then asking which outcomes answer to the result gives the same collection as combining the two EVENTS with a set operation. Those are two different routes, and nothing so far guarantees they arrive at the same place. So we are going to check. Start with the simplest of the three. Not A. The event that holds everything the experiment can still produce when A fails.

1:04One throw of a die, six outcomes, and take A to be a prime turning up: two, three, five. Then not A is one, four, six. Two things are true of that pair, and they are true together. Nothing is in both of them. And between them they are the whole list. That is not a happy accident of this example. Run it over every one of the sixty-four events this die has, and every single one of them shares nothing with its own complement and covers the list with it.

1:37And taking the complement twice gives you back exactly what you started with, on all sixty-four. There is one thing about the complement that is worth pinning down now, because it causes trouble later. It is taken against the outcome list, and against nothing else. Take the same collection, two, three and five. Inside a six-faced die its complement is one, four, six. Inside an eight-faced die the same collection has complement one, four, six, seven, eight.

2:14Same collection, different answer. So the complement depends on the experiment, and never on which other events you happen to be discussing at the time. Here is a complement written out in full, because writing one out is the fastest way to see what it is not. Three coins, tossed together, eight outcomes. Let A be exactly one tail turning up. Three outcomes answer to that: head-head-tail, head-tail-head, and tail-head-head.

2:46So not A holds the other five: three heads, head-tail-tail, tail-head-tail, tail-tail-head, and three tails. Three and five, adding to the eight, with nothing in both. Now look at what is sitting inside that complement together. Count the tails in each of those five. Nought tails, then two, two, two, then three. Three different values. And the one value that never appears is one, which is exactly the value the event was about.

3:18The complement is not a family with something in common. The all-heads outcome and the all-tails outcome are as unalike as this experiment allows, and they are in it together, joined by nothing except failing A. Take head-tail-tail. It is not in A, so on the throw that produces it the event fails and its complement occurs. Every one of the eight makes exactly one of the two happen, never both and never neither.

3:51Now two events at once. Back to one die, and two descriptions. A prime turning up: two, three, five. An odd number turning up: one, three, five. A prime OR an odd number is the union: everything in either one. One, two, three and five. Notice which outcomes are in there. Three and five are in BOTH descriptions, and they are in the union. Four outcomes, from a three and a three, because two of them were counted twice.

4:23Three plus three less two is four. And that is not arithmetic about this example. Take every pair of events on a five-outcome list, all one thousand and twenty-four of them, and the size of the union is always the two sizes less the size of the overlap. Not one exception. That sentence about the overlap looks like a technicality, and it is the condition the whole translation rests on.

4:55Ordinary speech very often uses OR to mean one or the other but not both. Try reading it that way here. A prime or an odd number would then name one and two, and nothing else. It would throw out three and five. But three is prime, and three is odd. It answers to the description. A collection that leaves it out is not the collection that description names, and now no set operation answers to the words at all.

5:26So how often does the difference bite? Run both readings over all one thousand and twenty-four pairs. They come apart on seven hundred and eighty-one of them, and agree on the other two hundred and forty-three. And the split is exact: they disagree on precisely the pairs whose overlap is not empty, and on no others. The inclusive reading is not a matter of taste. It is the one that makes the translation true.

5:58A AND B is the intersection: the outcomes answering to both descriptions. For the primes and the odds that is three and five. Check them one at a time. Three is prime and three is odd. Five is prime and five is odd. Both, for each of them, at once. And the intersection always sits inside each of the two events it came from, over every pair on the five-outcome list.

6:27There is one reading of AND that has to be stopped here. It does not mean the experiment is performed twice. One performance produces one outcome, and it is that single outcome that has to answer to both descriptions. Which is easiest to see on an experiment that looks like two things happening. Throw a die twice and write down both faces in order. Thirty-six outcomes, and every one of them is a single ordered pair: one outcome, not two.

7:01Let A be a six on the first throw. Six pairs answer to that: six-one, six-two, six-three, six-four, six-five, six-six. Let B be a total of at least eleven. The only totals that reach eleven are eleven and twelve. Two pairs give eleven and one pair gives twelve, so B holds three: five-six, six-five, six-six. Both descriptions hold together on two of them, six-five and six-six. So either description holds on seven, which is six and three less the two that were counted twice.

7:40And while we are here, the other two combinations. A six on the first but not a large total holds four pairs. A large total but not a six on the first holds one: five-six. So now the claim itself. There are two routes from a pair of descriptions to an event. Route one: combine the words, then walk the outcome list and keep whatever answers to the combined sentence.

8:09Route two: turn each description into its event first, then apply the set operation to the two events. Nothing said so far forces those to agree. So both routes were built separately and run against each other. Every event on a five-outcome list for NOT, which is thirty-two of them. Every pair of events for OR, for AND, and for BUT NOT, which is one thousand and twenty-four each. Disagreements: nought, nought, nought and nought.

8:42The connectives are the operations. That is a theorem about the translation, not a convention about notation. The fourth entry is A but not B, and it is the odd one out. It is the difference: what is left of A once whatever it shares with B is taken away. For the primes and the odds, that leaves two. Now watch it dissolve. An outcome survives A but not B when it is in A and out of B.

9:14Being out of B is being in not-B. So A but not B is A AND not B, which is an intersection and a complement, and nothing else. On the die, both routes give the single outcome two. Over all one thousand and twenty-four pairs, they never once part company. So the difference is not a fourth idea. It is a shorthand for two of the others. One warning about it, because it is the operation people reverse without noticing.

9:46A but not B is two. B but not A is one. Those are different sets, and their coming out the same size here is a coincidence of this example. Over every pair of events on the five-outcome list, the two ways round agree only when the two events are equal. Not once otherwise. And here is the other thing people get wrong about it. Taking A but not B does not throw away the whole of B.

10:16It throws away only what B shares with A. The odds have three members here; the difference discards two of them, three and five, and leaves the third alone, because one was never in the primes to begin with. Which also means the difference and the overlap put A back together: the single outcome two, with three and five, is the primes again. All four operations on one small example, so they can be seen together.

10:49One die. A is a prime turning up: two, three, five. B is an odd number turning up: one, three, five. A or B is one, two, three, five, and its size is four. A and B is three, five, and its size is two. A but not B is two, and its size is one. Not A is one, four, six, and its size is three. Two checks worth running on the board.

11:16Three and three less two is four, which is the union's size recovered from the other pieces. And the one-outcome difference together with the two-outcome overlap rebuilds A exactly. Every one of those numbers is a count of outcomes. Not one probability has been computed, and none is needed yet. Two small facts about numbers are carrying all four of those answers, and both of them are places people slip.

11:48One is not prime. The reason is a count: the divisors of one are just one itself, so it has exactly one divisor, and a prime has exactly two. Two has divisors one and two, exactly two of them, so two is prime. Now look at where those two numbers sit. One is odd and not prime, and that is the only reason B but not A has any member at all.

12:16Two is prime and not odd, and that is the only reason A but not B has one. Suppose you counted one as prime by mistake. Two of the four answers move: the overlap goes from two outcomes to three, and the complement goes from three outcomes to two. And two of them do not move at all. The union is still one, two, three, five, and the difference is still the single outcome two.

12:45Which is exactly why the mistake is hard to catch: half the answers still look right. Two more identities, and then a question worth asking. NOT of A or B is not-A AND not-B. NOT of A and B is not-A OR not-B. Read them in words and they are obvious. Failing to be in either one means failing to be in the first and failing to be in the second.

13:13Failing to be in both means failing at least one of them. Checked over all one thousand and twenty-four pairs, both of them, and nought exceptions either way. The interesting part is not that they are true. It is what they let you throw away. Turn the second identity round. If NOT of A and B is not-A or not-B, then A or B is NOT of not-A and not-B.

13:42The union, built out of a complement and an intersection. Turn the first one round the same way and the intersection is built out of a complement and a union. Both checked over every pair on the five-outcome list, nought exceptions. And the difference was already shorthand. So four operations were never four ideas. The complement, together with EITHER the union or the intersection, is the entire kit. Everything else in the algebra of events is spelled out of those two.

14:19Which raises the question that is never asked. Could you go further and manage with two of the operations that are not the complement? Keep OR and AND, take the complement away, and see how far two events get you. Start with A and B, apply the two operations in every way you can, then do it again to whatever you have made, and keep going until nothing new appears.

14:46From two events, that stops almost immediately. You reach A, you reach B, you reach their union and you reach their overlap, and that is all: never more than four collections, and closing it a second time adds nothing. It really does have to loop, by the way. Start from three events instead and one pass reaches seven collections while the closure reaches eight, and what the second pass adds is the whole outcome list.

15:17Now go looking for the complement inside that closure. Take a three-outcome experiment, so there are sixty-four pairs of events to try. On eight of the sixty-four the complement of the first event happens to be in there. On the other fifty-six it is not, and no amount of or-ing and and-ing will ever produce it. The die example is one of the fifty-six. Not a prime cannot be built from the primes and the odds by unions and intersections alone, however long you keep going.

15:53So the complement is doing work the other two cannot do. What has actually been established here is small and worth saying plainly. No new mathematics was introduced. Three English connectives were lined up against three set operations, and the line-up was checked rather than announced: over every event and every pair of events on a five-outcome list, both routes from words to a collection land on the same collection.

16:26That correspondence holds only if OR admits the outcomes lying in both, which is the one place ordinary speech would break it. Four operations turned out to be two: the difference is an intersection with a complement, and the union and the intersection are each other's under a complement. But two is where it stops. Without the complement, or and and between them cannot reach past four collections from two events, and the one you most want is usually not among them.

17:01And every quantity in all of that was a count of outcomes. The probabilities come next, and they will be added up across exactly these collections.