A Summer 2015 Mathematics A To Z: bijection


Bijection.

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.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

23 thoughts on “A Summer 2015 Mathematics A To Z: bijection”

    1. I think the only fair answer is “sometimes”. Occasionally a particular number will be part of the joke. Or an in-joke. For example, the number 47 turns up a lot in Star Trek because of a joke on the writing staff.

      But sometimes the cartoonist feels that, if there’s got to be some mathematical content, then it should be correct. Bill Amend of FoxTrot — a physics major, I feel the need to admit — does this a lot. (He’s also talked about the challenges of writing mathematics comics, and how that relates to teaching.) Some do that because it’s fun. Some because they feel it makes the joke stronger if the mathematics talk is authentic. Some, surely, because they learned this stuff so why not put it to use?

      And there are other cartoonists who just pick numbers because they look pretty, or they sound funny, or for other aesthetic reasons like that. And that’s fine also: comic strips are a form of art and aesthetic grounds have to count.

      But overall there just isn’t a universal rule.

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    1. It’s not far off. There’s a related term, “injection”, which would have made sense to put first if I were not doing this in the a-to-z order. (If I weren’t sticking with alphabetical order I’d have defined “function”, then “injection”, then “bijection”.) In an injective function everything in the domain is matched to something in the range, but it’s possible something in the range is missed.

      In a bijective function, everything in the domain matches exactly one thing in the range. So you can read it as two rules, one that matches everything in the domain to something in the range, and another rule that matches everything in the range to something in the domain. So in a way you can see it as a double injection.

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  1. Makes me think that ‘bijection’ could be a metaphor for a set of ‘perfect’ social grafts (i.e. old-fashion marriage), especially in engineered situations like those that favour monogamy, don’t allow divorce, and only recognize heterosexuality. Hmm, a Jane Austen novel? (Sorry… I recognized your sense of humour and couldn’t refrain.)

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    1. No need to apologize. Never fear that.

      I am interested in the metaphors that could be made out of bijection. Functions are all about pairing up things, one from the domain and one from the range. This does suggest parallels to social groupings, although I’m not sure how far the metaphor could be pushed before it stops being enlightening.

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          1. In the history of each metaphor, its birth probably brought shock and likely both amusement and disgust. Slowly, it aged to cliché and perhaps ceased to be recognized even as cliché (table’s leg). ‘Bijection’ is a really difficult term for metaphor; I have been thinking on it since reading your post. Putty and country dances might be the best, although one assumes too much experience with Jane Austen.

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              1. This might work: transposing music, when a tune is moved from one key to another, everything in the domain has a match in the range (that is to say, every note in the original key must be rewritten into the new one).

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                1. Oh, I think you have it, if not in transpositions then at least arranging a musical piece for a different instrument. A bijective function, among other things, lets you go from the range back to the domain. And some transpositions and arrangements let you reconstruct the original music perfectly.

                  But there are some functions, and some arrangements, that don’t allow that. For example, a song arranged for carousel organs will typically have a narrower range of notes available, and maybe less flexibility in time signatures. A skilled arranger can convert the song so that it sounds right on the new instrument, but you couldn’t reconstruct the original song perfectly from the carousel-organ version. The function matching the original song to the transposed/arranged version is not a bijective one.

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                  1. Cool. I was at the piano when the thought settled, but wondered if there would be difficulty transposing among instruments. Indeed, the range is not equal to all instruments. However, this discussion has helped illuminate the movement in bijection, as a kind of reciprocity. There must be a connection here to ratios and proportions, then?

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                    1. There might be a connection to ratios and proportions. I admit at this point I know so little music theory that I’d be wary of making a stupid mistake. I’ll make stupid mistakes, sure, but I do try to limit them to subjects I should know better about.

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                    2. I thank you for your many exchanges on the topic. My pet idea lately has been excess as discussed in cultural theory, and bijection feels like a strong example of efficiency and quite different, so it attracted me. Thanks again and have a great upcoming weekend.

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