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I am looking for suggestions about ways to introduce the preimage of a set under a function. My experience is that many students find it a confusing concept. The definition I use is as follows:

Suppose that $f:X\to Y$ is a function, and $A\subseteq Y$, then the preimage of $A$ under $f$ is $$f^{-1}[A]=\big\{x\in X:f(x)\in A\big\}\;.$$

Note that when teaching engineering students, I use the idea of a measuring device being a mapping from a physical state space $X$ to an observation space $Y$. You can then ask the question: Given a particular measurement/reading/observation, what physical states could have generated it? This approach is inspired by Steven M. LaValle's Sensing and Filtering: A Fresh Perspective Based on Preimages and Information Spaces.

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I would start more generally by asking how preimages and functions are related. Probably the example of $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $x \mapsto x^2$ would come up early. Does $f^{-1}[\mathbb{R}_{\geq 0}]$ correspond to a function? Why or why not? Then I'd build out with concrete examples of this nature... –  Benjamin Dickman Apr 30 at 16:53
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To me LaValle's approach looks like something a teacher could easily crash-and-burn with by spending too much time on jury-rigged explanations. The notation is probably the most challenging part in my opinion. As for the concept, I'd just say it's the least many (minimal set of) things in $X$ needed to obtain all the outputs represented by $A.$ Also, I wouldn't use $A$ for a subset of $Y.$ I'd use $A$ for a subset of $X$ and $B$ for a subset of $Y.$ –  Dave L Renfro Apr 30 at 16:53
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@DaveLRenfro You're wrong. $f^{-1}[A]$ is not minimal for non-injective functions $f$. –  Toscho Apr 30 at 20:12
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@Toscho: Ouch, I think that's the second time I've been corrected today! I wrote too quickly without thinking, and you're correct, of course, as the simplest examples show (a constant function). More correct would be to say that its all the things in $X$ with an output in $A$ and only those things. –  Dave L Renfro Apr 30 at 20:22
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The inverse image, under the "exam" mapping, of the set of sufficient grades will be the set of students who have passed the course. I think that's easy enough to understand. –  Marc van Leeuwen May 1 at 5:02
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I have thought about this a lot and, specifically, have wondered about the effects of terminology and notation. I'm not sure if there is a best answer, but here are some of my thoughts:

  • When I introduce image/preimage in an intro to proofs course, I avoid using the $f^{-1}$ notation, stressing for the students that we don't want to presume $f$ has an inverse. I understand this notation is fairly standard later on and elsewhere, but before a student really understands inverse functions, I think it's best to remove the potential opportunity to conflate these two completely different concepts that happen to share notation.
  • A professor of mine used "push-forward" and "pull-back" to refer to "image" and "preimage" and, the more I think about it, the more I really like those terms and want to use them (almost) exclusively myself. They really convey the operational aspect of the definitions: there's a function that does something, and we care about where it pushes domain elements (for the image) or where it pulls back to domain elements (for the preimage). Perhaps using these two terms in tandem with the standard terms will really tie the ideas together more coherently for students.
  • I like your example about measurements and observations, but I would recommend using even simpler examples, both of the arbitrary/abstract and applied varieties. Specifically, use some examples to show "surprising" behavior of the preimage operator. For instance, you can construct quickly (with students' help, even) a toy example to show that "the preimage of an image of a set is not necessarily equal to the starting set". Draw some schematic diagrams with dots and arrows. Then, draw a graph of a function $f:\mathbb{R}\to\mathbb{R}$ and ask about preimages of certain intervals on the $y$-axis: Where are all the $x$-values whose output "lands in" this set? (You could use this to introduce the idea from topology that a function is continuous when "the pull-back of an open set is open".)
  • Generally, as many examples as possible is, I think, the only way to really get used to this. Make sure some homework problems address these concepts. Make sure students have to "compute" preimages of certain sets, given a function, but also make sure they have to construct their own examples. Make them come up with a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f^{-1}[\mathbb{N}]=\varnothing$, say. Depending on the specific interests of your students (computer science, engineering, math) you can tailor some more examples/exercises.
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I treat relations first, and define functions as particular relations. The "inverse" I stress is just the transpose of the function as a relation in this. –  vonbrand Apr 30 at 17:09
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Regarding the $f^{-1}$ notation, I thought it was typical in 2nd and 3rd year (U.S.) transition courses to study relations in general before specializing to functions. When this is done, $f^{-1}$ always stands for the inverse relation, which may or may not be a function. –  Dave L Renfro Apr 30 at 17:12
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@vonbrand: Indeed, as do I. But remember that students come out of high school with an ingrained notion of what an inverse function is, using $f^{-1}$ all over the place. However, abstract relations, images, and preimages are very new to them. I think it's, understandably, incredibly difficult for them to simultaneously synthesize all these new ideas with their old understandings and notation. –  brendansullivan07 Apr 30 at 17:13
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@JW I learned the notation $f^-$ as a way to make a difference to the preknowledge $f^{-1}$. –  Toscho Apr 30 at 20:16
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@JW and Toscho: I've seen $f^{\rightarrow}$ for image sets and $f^{\leftarrow}$ for inverse image sets, both for functions and for relations. –  Dave L Renfro Apr 30 at 20:25
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Mapping between reality and observation demands a high level of scientific abstraction, that math students in such a course might not have. It's possibly easier to use physical (or other scientific) laws in the form of equations between values.

For example $U$-$I$-relationship in NTCs: Should this be a function $U(I)$ or a function $I(U)$? Physics can't give an answer, so Math should deal with both. But the relation is not injective, so how to deal about it?

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Do I assume correctly that by NTC you're referring to a Negative Temperature Coefficient thermistor, where $I$ is current and $U$ (or $V$) is voltage? –  J W Apr 30 at 20:31
    
@JW Yes, you assume correctly. –  Toscho May 1 at 9:24
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