What goes in, what comes out, and what was merely allowed to come out

0:00Here is a set. The numbers one through six. And here is a relation on it: keep a pair when the second entry is one more than the first. Walk the thirty-six pairs, ask that of each one, and five survive. One with two. Two with three. Three with four. Four with five. Five with six. That list is the relation. Now three sets get named. The things that open a pair. The things that close one. And the set we said we were working into.

0:33Two of those three are sitting in front of you already. The third one is not, and no amount of staring at the five pairs will produce it. That asymmetry is the whole of this video. Start with the two you can actually read off. Go along the list and collect every opening entry. One, two, three, four, five. Call that the domain. Now go along the same list and collect every closing entry.

1:03Two, three, four, five, six. Call that the range. Both of those were a walk down a written list. No decisions, no choices, nothing announced. Hand the same five pairs to anybody in the world and they get the same two sets. And notice something. Write one of the pairs down twice and nothing moves. Six entries written, five pairs held, and the domain still lists one just once. The domain is a set. It does not care how many times you wrote anything.

1:39Now the third one. When the relation was set up, a sentence was said out loud: this relation runs from the numbers one through six into the numbers one through six. That second announcement is the codomain. It is where the outputs are allowed to land. Allowed. Not where they do land. And it is not a measurement. It is a promise, made before any pair existed. Which means it cannot be recovered from the pairs, because the pairs were never asked.

2:12The definition that ties them together does not say the range equals the codomain. It says the range sits inside it. Containment, not equality. The rest of this video is about why that word was chosen so carefully. Look at what our five pairs actually did. The set had six numbers. The domain has five. Six opens nothing. Seven is not in the set, so six has nothing to be paired with.

2:44So the domain fell short of the first set. Now the other end. The range has five numbers too. One closes nothing, because zero is not in the set. So the range falls short of the codomain as well. One relation, and both shortfalls at once. Six lives in the codomain and in the range, but not in the domain. One lives in the domain and in the codomain, but not in the range.

3:13The domain and the range are the same size here, and they are still different sets. Each holds exactly one thing the other does not. Neither of them contains the other. Size tells you nothing about which is which. Here is a picture where the shortfall is impossible to miss. Left oval: nine, four, twenty-five. Right oval: five, three, two, one, minus two, minus three, minus five. Six arrows are drawn, and each one leaves a number and lands on a square root of it.

3:49Nine goes to three and to minus three. Four goes to two and minus two. Twenty-five to five and minus five. Collect the openings: all three left-hand numbers. The domain is the whole left oval. Collect the closings: six numbers. But the right oval holds seven. One sits there with nothing touching it. Why? Because to reach one, an arrow would have to leave a number whose square root is one.

4:20That number is one. And one is not in the left oval. So the range has six members, the codomain seven, and the containment is strict — visibly. That dark dot is not a mistake in the drawing. It is the ordinary situation. Put the first relation into the same kind of picture. Two ovals, one through six in each, five arrows. Six on the left is dark. One on the right is dark. One unused dot on each side.

4:54Now a third picture, to show what it looks like when nothing falls short. Left oval five, six, seven. Right oval three, four, five. Three arrows, each dropping its number by two. Every dot on the left is used. Every dot on the right is used. Domain is the whole left oval. Range is the whole right oval. And the codomain is that same right oval. Here the range and the codomain are the same set, and the containment is not strict.

5:29Which is exactly why this picture is a bad one to learn from. Two of the three sets have coincided, and a coincidence teaches you nothing about the difference. The domain and the range are still different, though: five, six, seven is not three, four, five. Now watch a domain fall short badly. Take the numbers one through fourteen. Keep a pair when the closing entry is three times the opening one.

6:00One goes to three. Two goes to six. Three goes to nine. Four goes to twelve. Five would go to fifteen. Fifteen is not in the set. So five opens nothing. Four pairs. That is the entire relation. The domain is one through four. The range is three, six, nine and twelve. And the codomain is all fourteen numbers, because that is what we announced. Ten of the fourteen open nothing at all, and every single one of them fails the same way: three times it lands past the end of the set.

6:39The announced set is more than three times the size of what actually arrives. Nobody is going to talk you out of that codomain by pointing at the pairs. It was declared. Now the opposite failure. A condition that refuses nothing. Take the whole numbers from minus four to four. Keep a pair when the difference of the two entries is a whole number. Eighty-one pairs to test. Eighty-one survive.

7:10The condition never once said no, because a whole number minus a whole number is always a whole number. Domain: the whole window. Range: the whole window. And here is the thing. You only found that out by testing it. Change the sentence a little and it is a completely different animal. Keep a pair when the first entry is bigger than the second. Same shape of sentence, same window. Thirty-six pairs survive, not eighty-one.

7:42And minus four opens nothing, because nothing in the window is smaller than it. So a condition being stated on a set tells you nothing about how much of that set ends up in the domain. Test it. Every time. One more, where everything comes out full. Take one, two, three, four and six. Keep a pair when the opening entry divides the closing one exactly. Twenty-five pairs in the product. Test them all.

8:14One divides all five of them. Two divides three of them. Three divides two. Four divides one. Six divides one. Twelve pairs. Now collect the openings. All five numbers appear. Collect the closings. All five appear there too. So the domain is the whole set, and the range is the whole set, and here they are equal to each other. Twelve pairs over a five-member domain. Far more pairs than openings.

8:47Must the domain and the range be the same size? Here they are. In the square-root picture the domain had three members and the range had six. Neither is a rule. Both are just what the pairs turned out to do. So what happens when somebody declares a different one? Back to the five pairs. Domain one to five, range two to six, codomain one to six. Now announce it again, into two through six instead.

9:18The pairs do not move. Not one of them. The domain is the same. The range is the same. And the codomain is a different set. The relation is untouched. What changed is what the notation records about it. Before, the range sat strictly inside the codomain. Now they are equal. Push further. Announce it into three through six. That breaks. Two closes one of the pairs, and two is not in that set, so this is no longer a relation into it at all.

9:54So the pairs do constrain the announced set. From below. It has to hold the range. It just does not have to stop there. Announce the same five pairs into the numbers up to a hundred and everything still works. A hundred allowed, five used. Given only the five pairs, how many different sets could have been announced? Confine ourselves to the numbers one through six. Two. One through seven: four.

10:25One through eight: eight. One through nine: sixteen. Every time the pool grows by one number, the count doubles. And all sixteen of those are genuinely different sets. Every one of them holds the whole range, so every one of them gives a working relation. And all sixteen read exactly the same domain, and exactly the same range. Sixteen answers, indistinguishable from the pairs. Exactly one of them is the range itself.

10:59If the codomain were computable, that count would be one. It is never one. Now the other direction. Hunt for the codomain of the three-times relation among the numbers one through nine. Zero. Nothing there can hold twelve. The pairs pin the codomain from underneath and leave it free above. That is what containment means. How special is falling short, really? Take three things on the left and three on the right. Nine possible pairs, and five hundred and twelve relations you could build.

11:37Check the containment on all of them. Five hundred and twelve out of five hundred and twelve. The range is inside the announced set every single time. That is not a coincidence, it is how the pairs were built. Now count the full ones. Three hundred and forty-three use every left-hand thing. Three hundred and forty-three reach every right-hand thing. Two hundred and sixty-five do both at once. And ninety-one do neither.

12:09So a hundred and sixty-nine of the five hundred and twelve leave something on the left unused. And a hundred and sixty-nine have a range strictly inside the codomain. Falling short is not the exception. It is just under a third of everything you can build. One last count, and it is a signpost. Of those five hundred and twelve, how many use every left-hand thing exactly once? Twenty-seven. Count it a second way to be sure: give the first thing one partner, the second thing one partner, the third thing one partner. Three choices each.

12:52Twenty-seven, and the two routes agree. So of the three hundred and forty-three that use everything on the left, three hundred and sixteen send something to two places at once. That tiny corner is where this is going next. A relation is allowed to skip things on the left, and allowed to send one thing to several places. Take away both freedoms and you get a much rarer object with a name of its own.

13:22But keep hold of the asymmetry, because it survives the whole journey. What goes in and what comes out, you read off the pairs. Where it was allowed to come out, somebody told you. And they could have told you sixteen different things.