History of the definition of Injective & Surjective Function

I'm a college student, just beginning to study Elementary Set Theory. In studying about the definition of injective and surjective function, out of curiosity, it came to my mind a question about the definition these functions:

1) How did they came up with such definition? How did it came about & why must it be defined that way? What is so special about injective & surjective function that makes them has to be defined in such a way?

To make clear the context of my question, here are the conditions of this question:

• "In short" injective functions are defined as: for every element in the codomain, there is at "most" one element that maps to it from the domain. For Surjective functions: for every element in the codomain, there is at "least" one element that maps to it from the domain.
• As you can see, i'm not seeking about what exactly the definition of an Injective or Surjective function is (a lot of sites provide that information just from googling), but rather about why is it defined that way? What makes it so special that it has to be given it's own terminology (injective & Surjective functions)?
• Lastly, i'm just a human so if something is not clear about this post, just let me know in the comment & i'll try to edit the post to make it clearer.

I don't know how the real story goes and this is just a rough speculation in trying to guess how it came about (it is possible to be something along this line but don't see it as the real truth though):

• The person who first coined these terms (surjective & injective functions) was, at first, trying to study about functions (in terms of set theory) & what conditions made them invertible. He observed that some functions are easily invertible ("bijective function") while some are not possible to have any kind of inverse. As he continue to study, he discovers that some functions can have inverses if certain conditions are met, as if they were "semi" invertible type of functions. He discovers that these conditions can be distuingish into two types: right invertible or left invertible. Later on he can distuingish that, only those function who maps with a specific type of pattern can have a left inverse (which he called the pattern "injective"). And a similar thing happened as he discovered the pattern for surjective functions. So in short my opinion is: is not like he wanted to define injective & surjective function like so, but it just happened that he "discovered it" in "that form". Hence, the definition of injective & Surjective function. But, again, this is just me trying to be Mr. Know it all. I hope I'm not being too rude. If anyone do know the real story, do share.

Regards,

According to this page on "earliest know uses of some mathematical words", the terms injective, surjective, and bijective were first introduced in Bourbaki's Théorie des ensembles, of 1954, page 80. The authors' motivations were to standardise terminology, stating :

Standard terms are badly needed for “one-to-one,” “onto” and “one-to-one onto”; will Bourbaki’s “injection,” “surjection” and “bijection” prove acceptable?"

(Note how the Bourbaki group refer to themselves in the third person.)

The title of the Bourbaki text supports your guess that these terms were first introduced in connection with the study of set theory.

The same site resource attributes the first usage of one-to-one correspondence (i.e., injection) to the Danish mathematician Zeuthen, in his Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un, of 1870.

The first use of "Onto", i.e., surjection, is credited to C. C. MacDuffee in his Introd. Abstract Algebra of 1940, shortly before Bourbaki's retreatment.

As to what makes these functions worthy of special attention, bijective functions are used to define equinumerousity in set theory. (A "two way" injection argument can also be used to identify equinumerous sets.) Bijective functions which are structure preserving are a used to identify isomorphic structures in algebra. And a host of other applications. In addition, injections, surjections, and bijections have interesting properties and are worth of study in their own right.

• This clarify things a little bit. To gain some insight, the question is: "but why aren't stardard terms "badly" needed for "many-to-one", "into" and "many-to-one into"? Clearly there is an implied objective / reason here. And from your answer, quite possibly it's for the study of Equinumerosity (and perhaps it is unlikely for the study of inversible functions like what I stated). Thanks. – Gin99 Jun 20 '16 at 20:36
• @Gin99 I was personally surprised by how recent this terminology is. I would take it that "many-to-one" and "into" are describable as "non-injective" and "non-surjective", so perhaps separate terms are not badly needed. A function has an inverse if and only if it is a bijection so the terms could be used interchangeably, depending on context. Although Bourbaki introduced these terms in relation to set theory, their applicability includes all mathematical theories featuring functions, which is pretty well everything. – Nick Jun 20 '16 at 20:55
• Thanks for the links. I am surprised that bijection/injection/surjection were used so early. I learned set theory using the one-to-one/onto terminology. In fact, I can't recall ever hearing the term bijection throughout my undergraduate degree in Math, which was several decades after Bourbaki. – Brad Nov 13 '20 at 8:38

Some complementary information :

• About prefixes "in" and "sur" : although "in" is very neutral and does not evoke especialy "one-to-one", "sur" means in French "onto" ("le chat est sur la table" = "the cat is on the table").

• The previous names in French were "univoque" (Middle age origin, contrary of "équivoque") used by mathematicians around 1900 for "injective" (https://www.cnrtl.fr/etymologie/univoque) and "bi-univoque" for "bijective".

• It must be said that injectivity and surjectivity are dual (in a certain sense) one of the other as examplified by "short exact sequences":

$$\displaystyle 0\to A\;{\xrightarrow {\ f\ }}\;B\;{\xrightarrow {\ g\ }}\;C\to 0$$

where $$f$$ is injective and $$g$$ is surjective, but in fact more than that : in the framework of categories, $$f$$ is a "monomorphism" and $$g$$ is an "epimorphism" : $$f$$ and $$g$$ operate on sets that have an algebraic/topological/geometrical structure, for example groups or vector spaces, or topological vector spaces, etc. Please note that prefixes "mono" and "epi" are the greek equivalent to "single/one" and "onto" resp.