On invertible functions and on functions in general

Once upon a time my University (Simon Fraser University) administered a survey about conditions for faculty employment. The survey was rather boring, but provided colleagues an opportunity to complain about lack of time for research, insufficient support from administration, large classes, poor ventilation, etc.  But this survey also had an interesting question: What do you like about your job at SFU? As this was towards the end of filling multiple pages, I responded with “an opportunity to ski on weekdays”. While Vancouverites know what I was referring to, the rest of Canadians may require an explanation. Vancouver’s three local mountains are overcrowded on weekends, so only those with flexible employment hours (or wealthy unemployed) can escape the crowds and enjoy the terrain on a weekday.

Jokes aside, I wish to provide a bit more serious answer about what I like about my job as a mathematics educator and researcher. It is an opportunity to extend my understanding of mathematics.  And there is always an opportunity to understand further the mathematics you already know, or think you know.

In the following I exemplify what I mean.

What functions are invertible? That is, what conditions are required for a function to have an inverse? I invite the readers to think of their own answer before reading further.

This rather simple question – where an answer is available for a high school student – resulted in a major disagreement between myself and a very respected and knowledgeable colleague. While I claimed furiously that in order to have an inverse, a function has to be one-to-one (injective) and onto (surjective), my colleague claimed passionately that only injectivity is required. Note that as of the moment of writing this note, AI agrees with me, but our disagreement took place before AI became the ultimate tie breaker, and there are multiple resources, including textbooks, that support either case.  So what’s the deal? Or, who is right?

This reminds me of an old parable of two fellows approaching a rabbi, seeking his ruling on their conflicting arguments. The rabbi listens carefully to the first fellow, and tells him, “You are right.” Then he listens to the second fellow and also tells him, “You are right”. A third fellow, who witnessed the event, approaches the rabbi in bewilderment and suggests, “They are presenting contradictory views, they cannot both be right!”  To which the rabbi responds, “Indeed, you are right, but…”.  

Back to conditions for function invertibility, the “right” view depends on the (implied) definition for a function. In fact, there are two, slightly different, but both accepted in the mathematical community definitions. One is the “ordered pairs definition”, that is, a function is a set of ordered pairs that is univalent. The meaning of univalence is that an element cannot appear in the first place in more than one ordered pair. Formally, if (a,b) \in  f  and (a,c) \in f  then b=c. Another is the “triple definition”, that is, a function as a triple (F, A, B), where A and B are sets and F is a univalent set of ordered pairs (x,y) where x \in A and y \in B. That is, for all x in A there exists a unique y in B (univalent) such that (x,y) is a member of F. The set A is the domain of the function, B is the codomain.

While the similarity is evident, a notable difference is explicit mention of domain and codomain in the “triple definition”. As such, adopting (even implicitly) the latter definition requires bijection (both injection and surjection) for function invertibility, while in adopting the “ordered pairs definition” injection is sufficient for the existence of inverse.

These issues are explored and nicely exemplified in Mirin, Milner, Wasserman, and Weber, K. (2020) (ask me for a copy if you cannot obtain it easily!). In fact, this article suggests to “see Zazkis & Marmur (2018), for a more thorough explanation”. 

That is where the disagreement, between the authors and the editor, regarding the requirement for invertibility started, and the parties respectfully agreed to disagree. Of interest, my informal investigation – which consisted of asking several colleagues-mathematicians – suggested that people mathematically-educated in North America tend to claim that injection is a sufficient requirement, while those mathematically-educated in Europe tend towards bijection. Do you confirm this observation?

Finally,

Are these two functions equivalent?

g: R→R, where g(x) = x²

h: R→[0, ∞), where h(x) = x²

References:

Mirin, A., Milner, F., Wasserman, N., & Weber, K. (2020). On two definitions of ‘function’. For the learning of mathematics, 41(3), 21-24.  

Zazkis R. & Marmur O. (2018). Groups to the rescue: responding to situations of contingency. In Wasserman, N. (Ed.) Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers, 363—381. Springer.