The name "surjection" makes some sense to me as "throwing onto/over the (whole) codomain, so as to cover it". I don't get the connection between the name "injection" and "hitting each element no more than once". Perhaps the "in" is privative rather than directional? If it just means "throwing into" then wouldn't any function be injective, since any function throws at least one element of its domain into the codomain?

Edit: Just discovered a duplicate here. No satisfactory answer though.

  • $\begingroup$ See also math.stackexchange.com/questions/202132/… $\endgroup$
    – lhf
    Oct 28 at 9:09
  • $\begingroup$ @lhf Thanks. I think a satisfactory answer to my question is given in a comment under the second answer: ' "injection" can be thought of as a synonym of "embedding", the smaller set is injected into the bigger one. '. $\endgroup$
    – mjc
    Oct 28 at 9:23
  • $\begingroup$ The Latin etimology from ĭnĭcĭo = throw in, introduce could be a factor and I would assume that surjection and bijection are actually derived from injection because they don't look like real words. $\endgroup$ Oct 28 at 9:24

The words "injective" and "surjective" are not that old. See here:

INJECTION, SURJECTION and BIJECTION. The OED records a use of injection by S. MacLane in the Bulletin of the American Mathematical Society (1950) and injective in Eilenberg and Steenrod in Foundations of Algebraic Topology (1952). However the family of terms is introduced on p. 80 of Nicholas Bourbaki’s Théorie des ensembles, Éléments de mathématique Première Partie, Livre I, Chapitres I, II (1954). Reviewing the book in the Journal of Symbolic Logic, R. O. Gandy (1959, p. 72) wrote:
Another useful function of Bourbaki’s treatise has been to standardise notation and terminology… Standard terms are badly needed for “one-to-one,” “onto” and “one-to-one onto”; will Bourbaki’s “injection,” “surjection” and “bijection” prove acceptable?
The terms did prove acceptable, even to mathematicians writing in English, and quickly became standard. For instance, all three terms are used in Jun-Ichi Igusa “Fibre Systems of Jacobian Varieties,” American Journal of Mathematics, 78, (1956), 171-199. (JSTOR search) The adjectival forms appear in C. Chevalley, Fundamental Concepts of Algebra (1956): “A homomorphism which is injective is called a monomorphism; a homomorphism which is surjective is called an epimorphism.” (OED)

You are certainly right that one cannot immediately derive from the word injective that distinct elements of the domain are mapped to distinct elements of the range. But in my opinion the colloquial use of "inject" suggests that distinct things remain distinct if they injected somewhere (for example molecules during an injection of a substance).

Anyway, we should accept that mathematical terminology is a specialist language allowing a lot of creativity. One should not try to overinterpret words used here.


OP answering own question.

I've now found this question or something like it posted in several places on the internet, including this site. As far as I can see, no one is clear on why 'injection' is a good name for what it describes. The closest I've seen to a satisfactory answer, in my view, is here:

"injection" can be thought of as a synonym of "embedding", the smaller set is injected into the bigger one.

But of course a smaller set can be the domain of a non-injective function. So the best connection I can see is that the domain of an injective function must be 'small' in the sense of no larger than its codomain, and the function then 'injects' the 'small' set into the 'large' set, and we sort of pretend that, by association, this implies what we actually want it to mean, namely that elements of the codomain don't get hit multiple times - even though they obviously could.


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