Time for another contest

Come up with the best mnemonic device for remembering which is injective and which is surjective.

53 Responses to “Time for another contest”

  1. Carl Says:

    When you inject heroin into your bloodstream, you have both heroin and blood in your bloodstream, but when you surject from your eyeballs, the blood comes out, but not all the heroin.


  2. Jay Says:

    Well, the one I’ve used for surjective, is “Sir, I’m on to you!” (although, I suppose I think of it as, “Sur, I’m onto you!”).

    Once I could remember that, I didn’t need an explicit mnemonic for the other.

  3. John Armstrong Says:

    Inclusions are injective, and “sur-” means “on”.. glossed as “covering”.

    Do people really have a problem with this? The only tricky bit I find is remembering which is right- and which is left-cancellable, and that has as much to do with notational conventions as anything else.

  4. Chris Says:

    Injection is what happens to a student reading Bourbaki. Surjection is what Bourbaki did writing their textbooks.

  5. namit Says:

    Injected spies would collect information about a subset of towns in the co-domain, but a “sur”ge of troops would leave out no town in the territory.

    And of course, since I have limited spies and too many towns in co-domain, I won’t have two spies covering the same town. However, I have numerous troops so I don’t mind having many of them on to one.

  6. Brent Says:

    I always think of the i’s in injective as 1’s and so you have two 1’s (1 to 1). For surjective I remember “unto” from the u and thus onto.

  7. Anatoly Says:

    Surjective is like a huge surge, a wave covering everything, not leaving anything uncovered. Injection is like a needle used for giving injections – thin and sharp, so 1-1.

  8. David Speyer Says:

    I find this one pretty easy. If A –> B is an injection, then A goes into B. “Sur” means “on” or “over” in French, and also shows up in English words like “surcoat”, “surcharge” and “surmount”. So just picture A on top of B, covering it completely.

    The one that did give me trouble for a while was epimorphism and monomorphism. Any clever tricks for this?

  9. Kevin Says:

    “Sir Jective hits everything with his sword.”

  10. Gareth Rees Says:

    Easy if you know a bit of Latin: the prefix in- means “into”, sur- (shortened from super) means “above” or “over”, and iacere is the verb “to throw”.

    I have trouble remembering which statistical errors are Type I and which are Type II.

  11. oz Says:

    what about convex vs. concave functions?
    (was always confused till i memorized that x^2 is convex)

  12. Buffalo Says:

    I always have trouble remembering these myself (does this mean I’m a bad computer scientist — yes).

    injective -> injustice…not everyone will get something mapped to it

    surjective -> surplus, surfeit…some will get too many things mapped to it

  13. Fred Hsu Says:

    When you’re 1:1 with someone you’re in their face, when you’re onto something you’re on its surface

  14. lf Says:

    I could never remember what ‘one-to-one’ actually meant until someone suggested I think of it as ‘two-to-two’.

  15. Mitch Says:

    into and onto is hard, too.

    What about:

    meet and join?

    codomain and range?

    natural, counting, and whole number?

    Here’s my try:

    “f is surjective, the range is covered.
    f is injective, no repeats discovered”

    If you say it over and over a lot, you’ll get over how bad it sounds.

  16. Erik Says:

    The mnemonic devices people come up with specifically to help themselves remember things aren’t always useful to everyone else. They often sound weird when repeated to others, but more importantly, they only need to cover the part that the creator has difficulty with. For example, if the words “injective” and “surjective” are already firmly planted in your head, even if you can’t remember which is which, then all you need is something to attach to one of the words, perhaps to just one part of one of the words.

    For me, I always just remembered what “injective” didn’t mean. The idea of an injection (as with a syringe or needle) didn’t seem compatible with the idea of getting everywhere, so that was enough for me. Also, I quickly latched on to the (for good reason) rarely used “into”/”onto” terminology, and the similarity between “into” and “injection” made it mostly a non-issue. But while this might work for some, it might not for others.

    Since the primary target of the mnemonic device is going to be students who are having difficulties, it’s also important that the mnemonic does not damage the student’s just-forming comprehension of the idea. That’s another potential problem with the metaphor of the injective function as injecting its domain into its codomain. It’s only an accurate metaphor in the sense that it doesn’t mesh with the concept of a surjective function very well.

    One way of overcoming this problem is to make the mnemonic device so abstract or bizarre that it doesn’t contain any conceptual content that could possibly interfere with useful metaphors. In this sense, “Sur, I’m on to you!” is fantastic. “Sir Jective hits everything with his sword” is also pretty good. Though oddly enough, I’ve encountered a couple students who have been thrown off by my using the word “hits” (without reference to knights of any name) to describe surjectivity, as in “this function hits everything in the codomain”.

    The ideal mnemonic device (and sorry, I don’t actually have one) would do this one better and actually help understanding. Something that would combine the ability to remember the words with a helpful metaphor. I’ve always used soda vending machines (with button presses as inputs and soda flavors as outputs) to describe the difference between injective and noninjective functions because it’s a context in which the students are very willing to accept two inputs mapping to the same output. It’s not useful for surjectivity, however, because that depends very strongly on a what codomain you specify. It also has some other weaknesses that I won’t go into here. I’m still looking for an ideal function metaphor in which something very natural corresponds to the idea of having a specified codomain, but I’m getting off-track here.

    I was hoping that by writing down what my ideal mnemonic device would be (funny, helpful for remembering the words as well as their associations, and helpful for comprehension (or at least not harmful)), I would be able to come up with one, but that was not to be today. Perhaps these thoughts will help someone else.

  17. Jadagul Says:

    David Speyer: I always found epimorphism and monomorphism easier, mainly because I took Greek. “mono” means one, which should be familiar (e.g. monotonous); that’s 1-1. “epi” is a Greek preposition meaning “on,” as in “epicycles” which are cycles on other cycles or “epidermis” which is the top layer of skin, the skin on all the other skin. So epi is onto.

  18. Erik Says:

    For others, I have help!

    “mono-” means one, so one-to-one is natural for “monomorphism”. “epi-” means all, so onto is natural for “epimorphism”. (Now if only this helped remember the category theoretic definitions, which I have to rebuild from scratch every time.)

    The symbol for meet looks like half an “M”. The symbol for join looks like a pointy “J” that is too long on the left.

    Forget the counting and whole numbers if you can. I’ve never seen them outside of high school math classes, and even then, they’re not important. (On a side note, for years, teachers told me that some people don’t include 0 as a natural number, but I’ve never actually encountered someone who did this. Have any of you?)

    If you imagine “range” as in “the target is within range of our weapons”, then it makes sense as the set of everything you can “hit” with the function. Helps to differentiate range from codomain, but not so much with the word “codomain” itself.

    If you can remember to think of things as pointing upward (i.e. the graph represents the upper edge of some solid object), then a function that forms a “cave” (or maybe just half of one) is concave. This is weak to some (in the same way that “righty-tighty, lefty-loosey” drove me bonkers as a kid) because it depends on an arbitrary direction. As far as polygons are concerned, this is a much better mnemonic device as we naturally fill in polygons as solid objects (or at least more naturally than filling “down” graphs).

  19. Samuel Says:

    I think what one should do in these scenarios is to memorize the most important of the terms, and remember that one.

    Injective is more important than surjective, because any function can be made surjective by just changing its codomain. (Technically, you could change the domain to make any function injective, but it’s much less natural and it might require the axiom of choice.)

    As for convex vs concave, I remembered that convexity is the more important concept, and so should be associated with a smiley face (the mouth of a smiling face is a convex function).

  20. boo Says:

    INjective goes INto.

    SURjective goes ONto.

    It helps if you know that “sur” is French for “on”.

    The construction of a mnemonic to relate SUR to ON is left to the reader as an exercise.

  21. boo Says:

    I have trouble remembering which statistical errors are Type I and which are Type II.

    Indeed. Talk about nonintuitive variable naming.

    This is why I always say “false positive” and “false negative”. You know exactly where you stand with those.

  22. svat Says:

    What about the question of whether “injective” and “surjective” are good words to use at all? They sound more impressive than “one-to-one” and “onto”, but if people need mnemonics to remember them, it’s probably a good idea to not use them, right?

    For convex and concave, I remember that a conVex function can look like a V, and a concave function can look like the entrance to a cave.

  23. Beetle B. Says:

    Injective: Mapping of A into B.

  24. bil gasarch Says:

    My Very Educated Mother Just Said Um- No Pluto.

    My favorite memory device.
    Due to Stephen Colbert.

  25. Mihai Says:

    Oh, injective and surjective are easy — just learn a Romance language. It took me until senior year at MIT to remember what “one to one” and “onto” mean, and the words still make no sense to me.

  26. Marius Gedminas Says:

    When I hear “one to one”, I always think people are talking about bijections.

  27. Mikael Vejdemo Johansson Says:

    I’m with John, Garreth, Mihai and several others on this – I don’t really see the need to construct mnemonics, since it’s reasonably obvious that one is into and the other is onto. For that, I find the englishified counterparts MUCH more confusing; onto is easy enough, but when does one-to-one mean injective, and when does it mean bijective??

    This is my response, also, to svat: injective, surjective, bijective have very clear, easily distinguishable meanings that are obvious with the smallest doses of Romance language knowledge (which will serve you well in other ways as well) – and not subject to the author-by-author randomness that, for instance, one-to-one is.

  28. Michael Luvaul Says:


    I now jam every chiffer into victorian estate.

    (makes a decent amount of sense as chiffer is number, victorian can be interpreted as proper and estate can be seen as domain)


    Sometimes ugly republicans jam every cardinal there is, violently ecclesiastical!

    (mostly a joke about dogmatic republican war mongers but it makes sense if you picture the republican covering every number in the domain)

  29. Michael Luvaul Says:

    Whoops, in both cases where I stated “domain” it should have been range or codomain. Sorry about that.

  30. tomate Says:

    Surjective is the useless one.

  31. Abel Says:

    Well, as many people have said, if you speak a language that comes from Latin, then the words actually make sense. That said, the way I actually remember it is just by knowing that x^2 is not injective, and x^3 is surjective.

    If I had to come with an mnemonic, well, you can give an injection to a cat. And you have Schrödinger’s cat, which we can say is in two places at the same time, which is what the elements of your range cannot do for injective functions. And then, surjective is the other one. It is not logical at all, but weird and stupid enough to make it easy to remember.

  32. James Says:

    What?! They mean exactly what they say. Injective means injective, and surjective means surjective. Just like dog means dog, and cat means cat.

  33. sirix Says:

    oz #11: I also have problems with these two and I started to mimick my friend who says convex downward (like x^2), and convex upward (like -x^2).

    True though that people sometimes fail to understand me; in that case I just wave hands to show which one do I mean:-)

  34. Douglas Knight Says:


  35. MattF Says:

    The key for me is to remember that it’s actually a threesome: injective, surjective, and bijective– With that, I can reconstruct the definitions instead of having to remember them.

  36. Darius Bacon Says:

    All these mentions of “injecting into” just don’t convey the idea to me somehow; since the actual idea is of distinct inputs producing distinct outputs, how about: injections preserve information.

  37. Johan Richter Says:

    I think injective-injection as well. If you inject a substance in your body the molecules aren’t going to end up in the same place, are they?

    I agree that one-to-one sounds like a bijection. Especially when you write it 1-1 where it seems symmetrical.

  38. culture-loving mathematician Says:

    This is another excellent illustration of why a knowledge of foreign languages should (still) be required to get a Ph.D. in mathematics.

    The idea of learning these terms without knowing (or, at the very least, simultaneously learning) their etymologies is…well, grotesque. (Like learning quantum mechanics the traditional way, except worse.)

    in-ject = “throw into”
    sur-ject = “throw onto”
    (cf. re-ject = “throw back”; pro-ject = “throw in-front-of”)

    As others have noted, the term “one-to-one” is awful. In addition to being kindergarten-sounding, it’s confusing: although a “function” is a “correspondence”, a “one-to-one function” (injection) is not the same as a “one-to-one correspondence” (bijection).

  39. david Says:

    I do not now why so many people suggest that knowing a Latin language makes it easier to remember which is which. Certainly not in Spanish, the words “injective” or “surjective” don’t evoke any meaning by themselves. (I also learned them by remembering the important one, namely injective.) But certainly they are much better than the “informal” counterparts: “onto” does not mean anything at all (how is this intuitive?), and “one to one” seems to be referring to a bijection.
    As for the poster claiming that knowing Latin should be a prerrequisite for earning a Ph.D in mathematics… well, that’s plain stupid. Sure it could help on certain occasions, but it’s a really unimportant thing.

  40. David Speyer Says:

    Thanks for the mono-/epi- help! So these are the Greek roots analogous to in-/sur-. I don’t know Greek but, now that I know they are supposed to work that way, it shouldn’t be hard to figure out which is which.

    I always remembered “e to the x is convex”. It rhymes, you see.

  41. culture-loving mathematician Says:

    “As for the poster claiming that knowing Latin should be a prerrequisite for earning a Ph.D in mathematics…well, that’s plain stupid.”

    I did not say Latin. I said “foreign languages”. Maybe you are unaware, but at least in the U.S. it is a tradition that mathematics graduate students are required to pass reading exams in two out of of three of French, German, and Russian — a tradition that, unfortunately, is in the process of dying out. My intention was to polemicize in favor of retaining this tradition. (I’m not the only one who thinks this. Paul Halmos also makes this argument in his memoir _I Want To Be A Mathematician_.)

    The language I was thinking of was mainly French, not Latin. (Indeed, from the point of view of Latin, “surjective” is a bastardization; it should have been “superjective” — cf. Spanish “suprayectiva”.)

  42. KaoriBlue Says:

    “Maybe you are unaware, but at least in the U.S. it is a tradition that mathematics graduate students are required to pass reading exams in two out of of three of French, German, and Russian…”

    That’s pretty neat, but where do they still do this?

  43. Raoul Ohio Says:

    1. I have had plenty of trouble learning certain opposite words, including AM and PM as a child. But I never had any trouble with injective and surjective. The following has always seemed like a natural progression:

    (1 to 1, onto) -> (into, onto) -> (injective, surjective)

    2. I first learned the words convex and concave with lenses, where cave concave is easy.

    But, how do these relate to functions? x^2 is concave if viewed from the top, and convex if viewed from below. Is there any reason why viewed from the top is the right way to think? I have always wondered about this. I suspect it is just bad terminology, although probably too late to get rid of now. What’s wrong with “curving upward” for convex.

    Does anyone know the back story on how this unfortunate term got ingrained into math?

    3. I have been busy and haven’t had a chance to read the “should children vote?” discussion. I bet a lot of nutcakes chimed in on that one. I look forward to finding out if Scott is one of them.

  44. Raoul Ohio Says:

    culture-loving mathematician:

    There are so many things it would be nice to learn, such as French, German, and Russian. Personally, I’d be happier with a good knowledge of combinatorics, probability, and statistics. You only have so much time to master stuff.

    For better or for worse, most science is done in English. Personally, I’m glad, because it is what I speak. And, it really ticks them off in France, where they used to outlaw reading scientific papers in English.

    The number of spoken languages in the world is rapidly shrinking. In a few decades most of the world will probably speak one of a half dozen languages. On one hand, that is a bummer, culture is being lost. I hope as much as possible is being recorded. But on the other hand, being a speaker of an oddball language is a huge disadvantage. English speakers all inherited it from family that at some time “got on board with the major language”, usually unwillingly. We are lucky because we are on board with a winner.

    It is a very interesting fact that such a huge part of the world speaks Indo-European languages, particularly English and Spanish. Presumably at some prehistoric time, some proto IE speakers got a lot of clout. A plausible guess is that at was the first crew, perhaps from Kazakhstan, that got in cahoots with horses as a military team, enabling them to take over a huge area. A few thousand years later, when technology was ripe to allow large scale colonization, speakers of two IE languages, English and Spanish, were the best at it.

    Halmos is a contradictory character. He is one of the best math writers ever. I own several of his books; they are great. But I think he is the guy who wrote “applied math is bad math”, or similar crap. I suspect he holds computer science beneath contempt. As anyone who has worked in both knows, pure math is semi recreational and lots easier than applied math. When you get to heaven, check out where Archimedes, Newton, Gauss, Euler, etc., have their tents pitched.

  45. culture-loving mathematician Says:

    “That’s pretty neat, but where do they still do this?”

    From what I hear it’s pretty much down to elite places like Princeton, Harvard, etc. Perhaps a few lesser schools still require one language.

    “Halmos is a contradictory character. He is one of the best math writers ever. I own several of his books; they are great. But I think he is the guy who wrote “applied math is bad math”, or similar crap.”

    Oh, dear. Come on — you can’t read an essay by title alone. Yes, Halmos (who is deceased) did write an article titled “Applied Mathematics is Bad Mathematics”. The title is deliberately provocative, in order to catch your attention. You should read the article; it doesn’t say what you think.

  46. John Armstrong Says:

    KaoriBlue: I had to pass language exams at the end of my second and third years at Yale.

  47. John Armstrong Says:

    Another side note that’s been bugging me about this whole thing: someone brought in “mono-” and “epi-” as synonyms to injective and surjective, which they aren’t.

    Well, maybe if you only ever work in the category of sets they are, but in general the monics properly contain the injections.

  48. Jadagul Says:

    KaoriBlue: Every program I applied to had the language requirement.

    John Armstrong: In undergrad I was taught that monomorphism and epimorphism were synonyms for injective and surjective respectively. I now know this isn’t true. On the other hand, it was true in every case I discussed in undergrad.

  49. Anonymous Says:

    That’s pretty neat, but where do they still do this?

    Harvard and Princeton both require two languages besides English for all math grad students; MIT, Chicago, and Berkeley require one.

    As anyone who has worked in both knows, pure math is semi recreational and lots easier than applied math.

    I find it difficult to believe anyone who could write this sentence has really done first-rate work in both.

  50. BPR Says:

    Sur-jective functions are sur-ly from having to ram something bigger into something smaller. (Where of course ‘bigger’ degenerates to the same size.)

    Injective is the other one. Or perhaps in-jective is ‘in-vertible-ish’.

  51. Johan Richter Says:

    I find it an interesting example of the way the mind works that no one has said anything about it being hard to recall what bijective means. It can hardly be that the term is more immediately transparent. (Bi means two. What has that to do with anything=?)

  52. culture-loving mathematician Says:

    “(Bi means two. What has that to do with anything=?)”

    “Two” as in “both”, i.e. both injectve and surjective.

  53. Paul Steckler Says:

    There’s a difference?

    — Paul