Can anyone give me a good explanation of how and why words surjection and injection came into use in mathematical community? What do they exactly mean? Who introduced them?

I have a feeling students prefer names like ''one-to-one'' when they first learn about functions because it tells them about the property in a simple and direct way. It takes some time and use of ''injective'' and ''surjective'' to start feeling natural when you are a beginner!

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    $\begingroup$ It's Bourbaki's fault. $\endgroup$
    – MJD
    Sep 25 '12 at 13:05

This is all speculation, but...

The French "injectif" is a natural choice, since we are injecting one set into another. The French word "sur" means "on" (as in "on top of"), making "surjectif" a portmanteau of sorts. I suspect the prefix "bi" has the same meaning in French as in English, and so "bijectif" refers to functions having the two properties of injectivity and surjectivity.

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    $\begingroup$ @ante.ceperic Hardly speculation... beyond French and back into Latin, this is exactly what the morphemes "in-", "sur-" and "ject" all mean. Part of what makes this terminology so good is that it is so suggestive and transparent. I suppose if the OP 's native tongue does not make use of such morphemes this would be a mystery... $\endgroup$
    – rschwieb
    Sep 25 '12 at 12:03
  • $\begingroup$ @rschwieb Thank you! This question with your comment explains everything. You are right, my native is Croatian (Slavic language) and I never learned Latin or French. So, root ''ject'' means ''to throw''? $\endgroup$ Sep 25 '12 at 12:47
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    $\begingroup$ @ante.ceperic Yes, exactly so. Probably the most common English cognate is "eject", which means to throw out. $\endgroup$
    – MJD
    Sep 25 '12 at 13:06
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    $\begingroup$ Or, famously, alea iacta est... $\endgroup$
    – Zhen Lin
    Sep 25 '12 at 13:15
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    $\begingroup$ @ante.ceperic Examples about in English because of all the Latin that comes through French. Surcharge, surfeit, surplus, surmount, eject, interject, reject, project, "in" has many, but I'm afraid of mixing them up with similar morphemes... I bet you'll be able to find plenty examples with "in" on your own. $\endgroup$
    – rschwieb
    Sep 25 '12 at 14:59

Wikipedia says that "the terminology was originally coined by the Bourbaki group".

The reason for using special words is precision: "one-to-one" is ambiguous, for some it means "injection", for others it means "bijection".

  • $\begingroup$ Thank you! I'm still interested in WHY those exact words, I doubt they chose them arbitrary. They must carry some sort of meaning, I believe. $\endgroup$ Sep 25 '12 at 11:25
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    $\begingroup$ Well, "injection" can be thought of as a synonym of "embedding", the smaller set is injected into the bigger one. "Surjection" (or "superjection" as my analysis teacher taught us) is also coming from latin, and the preposition "sur" (french version of latin "super") in meaning maximally reminds "onto". $\endgroup$
    – Berci
    Sep 25 '12 at 12:05

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