Tokens and symbols for injections, surjections, and bijections

[benjub]

Upon looking at #3053, I recalled how nightmarish the tokens and symbols for injection, surjection, and bijection are. Here is my suggestion:

• Tokens: >-->, -->>, >-->> (recall that the token for functions is -->).
• Latex: look at the code below and at its result in pdf:

\begin{gather*}
F \colon A \longrightarrow B\\
F \colon A \lhook\joinrel\longrightarrow B\\
F \colon A \longrightarrow\!\!\!\!\rightarrow B\\
F \colon A \lhook\joinrel\longrightarrow\!\!\!\!\rightarrow B
\end{gather*}

and the unicode versions of the above.

edit: here is a screenshot:

[benjub]

Here is a more detailed account:

[jkingdon]

I'm not a fan of this change.

Now, perhaps I'm biased because I learned this with the 1-1 and onto terminology rather than the injection and surjection terminology,[1] but I never can remember which kind of arrow corresponds with which concept - I always have to look it up.

I suppose it is a case of "The challenge of varying mathematical conventions" from https://us.metamath.org/mpeuni/conventions.html which does illustrate the problem but doesn't necessarily tell us how to resolve it. I can make an elaborate plea that the 1-1 and onto notation (with the text over the arrows) is far easier for beginners but even an assertion like that is hard to prove or disprove.

[1] it is long enough ago that I'm not sure I can find the name of the set theory text, but it wasn't metamath, it was an intro set theory textbook.

[jkingdon]

According to https://us.metamath.org/mpeuni/df-f1o.html the notation with the 1-1 and onto above/below the arrow is from [TakeutiZaring]

[digama0]

I'm not a fan of this change.

Now, perhaps I'm biased because I learned this with the 1-1 and onto terminology rather than the injection and surjection terminology,[1] but I never can remember which kind of arrow corresponds with which concept - I always have to look it up.

For the record, I believe this is also Norm's stated reason for using this syntax instead of those arrow variations, which he was aware of.

[benjub]

Ok. Then less work for me, I guess.

Just for the record: a quote from https://mathshistory.st-andrews.ac.uk/Miller/mathword/i/ (note the dates!).

... 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. ...

[I left the extra "h" in "Nicholas".]

A quick search from the arXiv[math], searching in titles:

• "injective": 775
• "one-to-one": 20
• "surjective": 209
• "onto function" or "onto mapping": 1
• "bijective": 441
• "one-to-one" and "onto": 0

There are a few false positives and negatives; "onto" gives 177 results, which is still less than for "surjective", even though "onto" is most of the time used as a simple preposition. Note that the search also matches words of the same family (e.g. injective, injection, injectivity), and indeed the grammatical regularity of these words is a pro. Finally, "injective" has a lot of hits since it is used not only for functions but for e.g. "injective module", "injective resolution"... but all these terms were coined because of the "injective" of "injective function" anyway. If one has time, one could do a similar search for books in a given representative series, say all the GTM books, and see how many of them use one-to-one-onto and how many use in/sur/bijective.

[david-a-wheeler]

A quick search from the arXiv[math], searching in titles ...

Good thing to check, but there are different kinds fo hyphens (and someone could omit them). Might that affect the result?