The one-output rule that promotes a relation to a function

0:00A function is a relation. Just a special one. And the sentence that says which ones are special makes two demands, not one. Most people hear the second and walk away. Here they are, separately. One: every element of the first set has to be sent somewhere. Two: no element may be sent to two different places. Two demands, and a relation can pass either one while failing the other.

0:29That is not a theoretical worry. There are relations that do exactly that, and we are going to look at one of each. Neither demand is optional, and neither one implies the other. A relation is a list of pairs, so both demands become tests you can run on a list. First test. Take the first set. Go through it one member at a time, and ask whether that member opens a pair.

0:58If any of them opens nothing, the first demand fails. Second test. Take the pairs themselves. Compare every one against every other. If two of them start the same way and end differently, the second demand fails. Notice what the second test does not say. Two pairs that start the same way and end the same way are fine. They are the same pair written twice. Write a pair down twice and nothing changes at all. Four entries written, three pairs held, and still nothing is sent to two places.

1:35The forbidden thing is not repetition. It is disagreement. Two tests, run separately, on the same list. First relation. The numbers one through six, and the pairs where the second entry is one more than the first. One with two, two with three, three with four, four with five, five with six. Run the second test. Compare all five pairs against each other. No two of them start the same way. Nothing is sent to two places.

2:08The second demand is met perfectly. Now the first test. Walk the first set. One opens a pair. Two, three, four, five, all open pairs. Six opens nothing. There is no seven in the set for it to go to. So the first demand fails, and this is not a function. And here is where almost everybody gets the direction wrong. Six is not the problem because six is missing from the outputs. Six is an output. Five is sent to it.

2:42Six is a problem because it is not an input. Ask what six is sent to and there is no answer to give. Shrink the first set to one through five and the very same pairs become a function. Nothing about the pairs changed. What changed is what we promised to cover. Second relation, and it fails the other way round. Three numbers on the left: nine, four and twenty-five. Seven on the right.

3:13Six arrows, each one landing on a square root of the number it left. Run the first test. Nine opens a pair. Four opens a pair. Twenty-five opens a pair. Every member of the first set is sent somewhere. The first demand is met. Now the second test. Nine goes to three. And nine also goes to minus three. Same start, different ends. So the second demand fails, and all three of the left-hand numbers fail it.

3:46This one is not a function either, for the opposite reason. And notice what that costs you. For the first relation, five was sent to six, and you could say the image of five is six. One thing. Here, ask what the image of nine is and the question has no answer. There are two candidates and nothing to choose between them. The word the only becomes legitimate after the second test has passed.

4:16That is what the definition is buying. So the second demand forbids one input going to two outputs. People flip it round and think two inputs must not share an output. That is not what it says, and it is worth seeing the extreme case. Six inputs: two, five, eight, eleven, fourteen, seventeen. Send every single one of them to the number one. First test: all six open a pair. Passes.

4:48Second test: no input opens two pairs. Passes. It is a function. Six inputs, one output, and nothing wrong with it at all. Now reverse every pair. One with two, one with five, one with eight, and so on. Now the number one opens six pairs, going to six different places. The second demand fails at once. Same six pairs, turned round, and a function became something that is not one.

5:19Many inputs to one output survives. One input to many does not. The arrow has a direction and so does the rule. Three lists. Run the two tests on each. First list: two with one, three with one, four with two. Openings are two, three and four. All different, so nothing is sent twice. And every opening is in the first set, which is those three numbers. A function. Its range is one and two.

5:50Notice that one is reached from both two and three, and that is allowed. Second list: two with two, two with four, three with three, four with four. Two opens twice, and it goes to two and to four. Different ends, same start. Not a function, and two is the only culprit. Third list: one with two, two with three, and so on up to six with seven. Six pairs, six different openings, all of one through six.

6:20A function. Domain one to six, range two to seven. And in this one nothing is reached twice either, which is extra, not required. So how special is this? Take three things on the left and three on the right. Nine possible pairs. Five hundred and twelve relations. Run both tests on all five hundred and twelve and sort them into four piles. Passing both: twenty-seven. Passing the first and failing the second — covering everything but sending something twice: three hundred and sixteen.

7:00Passing the second and failing the first — sending nothing twice but leaving something out: thirty-seven. Failing both: a hundred and thirty-two. Every one of those four piles has something in it. That is the proof that the two demands are independent. If the first implied the second, one pile would be empty. If the second implied the first, another would be. Neither is empty. And twenty-seven out of five hundred and twelve is about one in nineteen.

7:35A function is a rare kind of relation, and it is rare because it has to pass two tests, not one. Stay with those twenty-seven for a moment. How many of them send two different inputs to the same place? Twenty-one. So repeated outputs are not the exception among functions. They are the ordinary case. The other six never repeat an output. Now reverse every pair of every one of the twenty-seven, and test again.

8:08How many are still functions? Six. And they are exactly the six that never repeated an output. Twenty-one of the twenty-seven stop being functions the moment you turn them round. That is the asymmetry, counted. Being a function is a statement about the direction you are reading the pairs in. None of this needed the first set to be small. Take the counting numbers, all of them, and send each one to its double.

8:41You cannot write the list out. You can still run both tests. First test: does every counting number have a double? Yes. Second test: does any counting number have two different doubles? No. A function. Its outputs are the even numbers, and only the even numbers. Nothing odd is ever reached. So the range sits strictly inside the counting numbers, which is fine — nothing in either demand says a function must reach everything.

9:14And now the notation is safe to use. Write it with an arrow: from the counting numbers to the counting numbers. What an input is sent to is called its image. Seven has the image fourteen. Going the other way, seven is a preimage of fourteen. Note the words. The image, because there is exactly one. A preimage, because there might be several. The definition earns you the word the.

9:44Now the sharpest case of all, and it is two rules glued together. Take the whole numbers from nought to ten. Square the input up to three. Triple it from three to ten. The two stretches overlap at exactly one input: three. So what is three sent to? Three squared is nine. Three times three is nine. The two rules agree. One answer, so three is sent to one place.

10:12First test passes, second test passes. A function. Now change one number. Square the input up to two. Triple it from two to ten. Everything else is identical. Two squared is four. Three times two is six. Four against six. The input two is sent to two different numbers. Not a function. And nothing else separates the two definitions. Being written with formulas is not what makes something a function. The formulas have to agree where they meet.

10:49That is two examples. But the split point was a choice, so let us make it the question. Square up to the split, triple from the split to ten. There are eleven whole numbers you could split at. Run the two tests on all eleven. Two of them give a function. Nought and three. At nought, the squaring stretch is a single point, and nought squared and three times nought are both nought.

11:16At three, nine and nine. Everywhere else, the two rules disagree at the split. And when they disagree, exactly one input is sent to two places — the split itself. Not a whole stretch. One number. Nine of the eleven fail, and each one fails at a single point. That is how narrow the failure is, and how completely it disqualifies the whole thing. One more, and here the counterexample has to be found rather than quoted.

11:50Take pairs where the first entry is a product of two whole numbers and the second is their sum. Is that a function? Try six. Six is two times three, and two plus three is five. But six is also one times six, and one plus six is seven. Five and seven. Same start, different ends. So no. And you found that by looking, not by being told. Now step back and count. Letting both whole numbers run from minus six to six, thirty-seven different products come up.

12:27Thirty-two of them are sent to more than one place. Only five are not. And those five are interesting. Minus one, minus nine, minus sixteen, minus twenty-five and minus thirty-six. Every one of them is a number times its own negative, and that is the only way to make it here, so every one of them is sent to nought. The ambiguity is the rule. Being sent to one place is the exception.

12:59Two more names, and they are not the same name. A function is called real valued when its outputs are all real numbers. That is a condition on what comes out. A function is called real when it is real valued and its inputs are real numbers too. That is a second condition, on what goes in. So the second name is the narrower one. Now, a trap. Try to separate them with a function on the counting numbers and you will fail.

13:32The counting numbers are real numbers. Doubling and adding one, on the counting numbers, satisfies both conditions. It is real valued, and it is real. To separate the two you need inputs that are not numbers at all. Send three names — Amara, Bruno and Chloe — each to the number five. That is a function. Its outputs are real numbers, so it is real valued. Its inputs are names. So it is not a real function.

14:05One condition met, the other not. That is what the second name is doing. And now the reason any of this matters, which is that you can finally just use the thing. Take the counting numbers, double the input and add one. One goes to three. Two goes to five. Three, seven. Four, nine. Five, eleven. Six, thirteen. Seven, fifteen. A table, and every entry in it has exactly one right answer.

14:37That is the whole payoff. You can ask what an input is sent to and expect an answer. Try another. Double the input and take away five. At nought, minus five. At seven, nine. At minus three, minus eleven. Negative inputs are fine. Nothing in either demand mentions signs. Last one, with units on it. Nine fifths of the input, plus thirty-two. Degrees in, degrees out. Nought goes to thirty-two. Minus ten goes to fourteen.

15:12Twenty-eight goes to eighty-two point four — an output that is not a whole number, which nothing forbids. And you can run it backwards. Which input gives two hundred and twelve? Exactly one does, and it is a hundred. Exactly one, because that is what the second demand guaranteed. Every input, one answer.