# How did “one-to-one” come to be used to refer to injective functions?

I have always had a hard time explaining to my students the term one-to-one. After making sure my students understand "in", "sur" and "bi", the Bourbaki terms, injective, surjective and bijective make sense to them. But to me, one-to-one sounds like it should mean bijective, not injective. I sometimes tell my students that injective should be called two-to-two, meaning that two different elements are mapped to two different elements.

ONE-TO-ONE CORRESPONDENCE is found in H. G. Zeuthen, "Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un," C. R. LXX. 742. (1870).

One-to-one correspondence is found in English in 1873 in Proc. Lond. Math. Soc. IV. 252: "The equations .. being supposed to establish a 'one-to-one' correspondence between the two integral spaces." In his Principles of Mathematics Bertrand Russell (1903, p. 113) states, “Two classes have the same number...when their terms can be correlated one to one, so that any one term of either corresponds to one and one only term of the other.” (OED).

It sounds to me like Russel is talking about a bijection. Who and when started using "one-to-one" to mean injection?

• See the "subtle" difference between one-to-one function and one-to-one correspondence: In the second case, we have an inverse which is also one-to-one. – Mauro ALLEGRANZA Mar 19 '16 at 12:53
• "A one-to-one function from $A$ onto $B$ is called a one-to-one correspondence between $A$ and $B$." Herbert Enderton, Elemets of set theory (1977), page 129. – Mauro ALLEGRANZA Mar 19 '16 at 12:57
• I am guessing that "one-to-one" refers to every one point in the domain remaining one (alone) when mapped, no two points come together. This is particularly visual when two bubbles are drawn for domain and codomain with connecting lines representing the map. – Conifold Mar 20 '16 at 21:43
• Yes, I know what you mean, but I come across students who confuse one-to-one sounds with the definition of a function. You have to interpret one-to-one as meaning one-to-one, starting with a point in the image. – Helmer.Aslaksen Mar 21 '16 at 18:12