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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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    It's Bourbaki's fault.2012-09-25

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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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    @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.2012-09-25
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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".

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    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".2012-09-25