To explain this second term in my mathematical A to Z challenge I have to describe yet another term. That’s function. A non-mathematician’s idea a function is something like “a line with a bunch of x’s in it, and maybe also a cosine or something”. That’s fair enough, although it’s a bit like defining chemistry as “mixing together colored, bubbling liquids until something explodes”.
By a function a mathematician means a rule describing how to pair up things found in one set, called the domain, with the things found in another set, called the range. The domain and the range can be collections of anything. They can be counting numbers, real numbers, letters, shoes, even collections of numbers or sets of shoes. They can be the same kinds of thing. They can be different kinds of thing.
For example, a function might describe where on the Turnpike a car has gotten to. Its domain may be the time elapsed since travel began. Its range that’s the mile marker the car has reached. The rule connecting the time elapsed to the mile marker reached might something like “55 times the time elapsed in minutes, plus 100”. In shorthand we might say if t is the time elapsed in minutes, and x is the mile marker reached, then “x = 55 * t + 100”.
Domains and ranges and rules can be anything. So we can come up with functions that might seem pretty abstract. “The domain is the calendar year, and the range is the set of shoes a person wears, and the rule connecting them is to match the calendar year up to whatever pair of shoes the person wore on the 27th of May of that year” is a legitimate if phenomenally useless function.
Since mathematicians work with numbers so much, numbers are the most common domains and ranges for functions. They’re so common that often we’ll simply describe the rule. We take it for granted that both the domain and the range are the set of real numbers. Context might require we vary that a little. Sometimes it’s obvious we mean just the integer number, or just the counting numbers. Sometimes it’s obvious we mean complex-valued numbers. And sometimes a rule doesn’t make sense for every possible number, by calling for a division by zero or something similarly offensive. Then we take the domain to be “what we assumed before, only without the numbers that would make it not make sense”.
In higher mathematics we might get to where the domain or the range have to be sets of spaces, or matrices, or other functions, or other mathematical constructs. At this point mathematics majors realize they don’t have any idea what’s going on anymore. This stage passes. Or they transfer, I assume to electrical engineering or something.
Now, a bijection is a function that has two particular properties. The first is called being “one-to-one”. I’m almost afraid to define that term. It might be easier left as it is. I’ll try anyway. In a one-to-one function everything in the domain gets matched to one thing in the range. Furthermore, there’s nothing in the range that’s matched with more than one thing in the domain.
Let me give an example. Let my domain be the numbers from -3 to 3, and let me use x as a name for numbers in the domain. Let my range be the numbers from -27 to 27, and let me use y as a name for numbers in the range. Then the function that matches x with y = x^3 is a one-to-one function. The number 8 in the range matches with the number 2 in the domain, and no other. The number -1 in the range matches with the number -1 in the domain, and no other. And so on.
However, the function — with the same domain and range — that matches x with y = x^2 is not one-to-one. The number 4 in the range matches with both -2 and +2 in the domain.
Being one-to-one is a property of the function’s rule, yes. It’s also a property of the domain and the range. All three come together to make a function one-to-one or else not one-to-one. For example, if the domain is the numbers from 0 to 3 and the range is the numbers from -27 to 27, then the rule matching x in the domain to y = x^2 in the range is one-to-one. The number 4 in the range matches only to the number 2 in this domain. For this function, there’s no such thing as minus two in the domain.
The second property that a function has to have in order to be a bijection is that it must be “onto”. By “onto” we mean that everything in the range is matched up to something in the domain. It might be matched to two, or three, or infinitely many things in the domain. “Onto” means everything in the range has to be matched with something, somewhere, in the domain.
To go back to my examples, with the domain of -3 to 3 and the range of -27 to 27. The function with the rule matching x to y = x^3 is onto. Every number y from -27 to 27 is matched with something in the domain. Specifically it’s matched with the cube root of y.
But the function with the same domain and range, but the rule matching x to y = x^2, is not onto. Numbers y from -9 to 9 are matched to something in the domain, fine. But there’s no x between -3 and 3 for which x^2 is 16, or 25, or -10.
As with being one-to-one, being onto is a property of the rule, the domain, and the range all together. The function with the domain of numbers -3 to 3, and the range of the numbers 0 to 9, and the rule of y = x^2 is onto again.
Being one-to-one doesn’t require a function to be onto. And being onto doesn’t require a function be one-to-one. If a function has both the properties of being one-to-one and being onto simultaneously, then we call it a bijection.
Bijections can be quite nice to have. If you think of a function as taking an input and giving an output, then bijections are functions you always know how to undo, for example. That’s reassuring. But let me describe one of the ways mathematicians see bijections.
Imagine the domain as a set, as a kind of amoebic blob of stretchable putty sitting on the table. Maybe think of it as cookie dough, if it’s closer to snack time. And imagine the range as an outlined shape somewhere else on the table. You can think of a function as picking up this domain-putty-blob and dropping it somewhere within the range. The blob might get stretched out in the picking-up and dropping-down; that’s fine. It might get squashed down, or folded over. It might even rip. It might not fill the whole of the outlined range.
But if the function is a bijection, then it describes a way to pick up the putty-blob of the domain and drop it down so it exactly fills up the range. There’s no folding over, there’s no missed spots in the range. It fits exactly from that start to that end.
If you drew a figure on the putty-blob before picking it up, the figure would still be there after the bijection drops it into the range. The figure might be distorted, possibly terribly distorted. But it’ll still be there. If you can imagine this, you are ready for some serious, professional-level mathematics work.