Does anyone know how the definitions for onto and into map to the spoken language definitions of the words? I compared the Bourbaki definitions to these words and have a suspicion that the German word thrawn, meaning "twisted" may clarify this.


Injective = every element of the co-domain is mapped to by at most one element of the domain. 1-1. May be Into.

Surjective = every element of the co-domain is mapped to by at least one element of the domain. Is Onto.


Into: expressing movement or action with the result that something becomes enclosed or surrounding by something else.

In,En: to restrict, on all sides completely. Negative force Latin.

To: proposition indicating relation to another word, to place, before, indicates direction towards something

Onto: moving to a location on (the surface of something), moving aboard with intention of traveling in.

On: Being (Greek), above and in contact, (German) directly supported by another, by, near, close, next to, followed by another time point.


  • In: negative force
  • jacere: throw (Latin)
  • ive: relating, belonging, tending to


  • Sur: over, above, up
  • jacere: throw (Latin)
  • ive: relating, belonging, tending to

I'm also wondering if the translation of jacere really means "twisted, distorted" from the German word thrawn. If thats the case, I can see how the word injection really means "not a twisted relationship from above" and surjection means "a twisted relationship from above". Where from above represents the function F mapping domain to co-domain or the reverse. This would imply injection is a "1-1 mapping preserved by the function"" and surjection is a "1-1 mapping not preserved by the function". From this interpretation, it seems as if there is some topological relation. Into keeps the form of the domain. Onto has twisted form/surface because of the possible many to one relationship from domain to co-domain.

Into Example $F$ maps $A$ into $B$

$F: A \rightarrow B$ and $F \subseteq B$ and $F=A$

Onto Example $F$ maps $A$ onto $B$

$F: A \rightarrow B$ and $F=B$

Appreciate anyone's insight here. I'm trying to clear up this language because it makes the math easier.


  • 2
    $\begingroup$ I think Into=injective is wrong in English. $\endgroup$ Sep 1, 2017 at 20:51
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    $\begingroup$ While I do think this is an interesting question, I'm afraid it's off-topic here. This is an etymology question where the historical context plays no role. (Or does it?) $\endgroup$
    – Ben
    Sep 2, 2017 at 20:54
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    $\begingroup$ I do not think so... The French Bourbaki has injection from Latin iniectio with the "usul" meaning of "inserting" and surjectionthat sounds a neologism adapted from the previous by analogy, due to the def of an "application de $E$ sur $F$" in the case when all elements of $F$ are "covered" by the map. $\endgroup$ Sep 3, 2017 at 15:03
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    $\begingroup$ This whole question is a jumble of nonsense. There is no German word remotely resembling "thrawn". Latin "in" has nothing to do with "negative force". The English preposition "on" has nothing to do with the Greek participle ων. Need I go on? $\endgroup$
    – fdb
    Sep 6, 2017 at 18:22
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    $\begingroup$ I do not understand what the question "how the definitions for onto and into map to the spoken language definitions of the words?" means. Spoken language words have no definitions, dictionaries simply try to describe common usage, and terms typically do not reflect non-mathematical usage in fine detail. So what exactly is being asked? $\endgroup$
    – Conifold
    Sep 6, 2017 at 23:52


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