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I was aware that students find the definition of function too abstract and thus find it difficult. However, I thought, once you understand functions, the concept of injective and surjective functions are easy. True to my belief students were able to grasp the concept of surjective functions very easily. On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say "a function is one-one if every element in the domain has a unique image". I am unable to help my students as I do not understand the difficulty they are facing. And this makes me feel helpless and frustrated. Do students in general find the concept difficult? If yes, what makes the concept difficult?

I could think of a possible explanation, but I was not so sure. The definition contains a $p\implies q$ statement and I know many students find logic difficult. So, the students might be finding injective functions difficult for similar reasons.

Finally, if the concept is not generally difficult for students, then I might be doing something wrong. What are the common mistakes teachers make while teaching the concept? Knowing this would help me steer off the common pitfalls.

Thanks in advance for any help you can provide.

Edit 1: These students are mathematics major students fresh out of school.

Edit 2: I accepted an answer as the answer really flushed out the line of thought I had in mind. However, I would still love to see more answers and suggestions on how to make the concept easier for the student.

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    $\begingroup$ When teaching this concept to college algebra and precalculus students (nearly all students were not even science majors, let alone math majors) I discussed it graphically. They all knew the vertical line test for a function, so I would introduced the horizontal line test to check whether the function was one-to-one (the fancy word "injective" was never mentioned!), which you might try. Incidentally, I made this name up around 1984 when teaching college algebra and precalculus courses. (continued) $\endgroup$ – Dave L Renfro Nov 26 '20 at 15:57
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    $\begingroup$ It wasn't in any textbooks I saw then, but by the 1990s the term started to show up in college algebra and precalculus textbooks, and even more so since 2000. However, by then I'd also found the term (browsing library books) used in a few books published before 1984. Also, I seriously doubt my usage had anything to do with the later usage as it's such a natural sibling to "vertical line test" that I am sure (now, and also years ago) that others must have also thought of the term, plus I probably wouldn't have mentioned it on the internet or in email to anyone until at least the mid 1990s. $\endgroup$ – Dave L Renfro Nov 26 '20 at 16:00
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    $\begingroup$ You have identified a concept in the "zone of proximal development" for you students. This is good teaching. You have identified something that is not too easy, and not too difficult. $\endgroup$ – user52817 Nov 26 '20 at 16:38
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    $\begingroup$ The answers posted already have some good ideas, so I'll just mention another here: the notion of injectivity and the notion of just being a well-defined function are very similar in that they both contrast "one" with "many". The difference is the order. A function is well-defined when it is one-to-one, NOT one-to-many. But a function is injective when it is one-to-one, NOT many-to-one. This similarity may contribute to the swirl of confusion in students' minds and, as others have pointed out, this may just be an inherent, perennial difficulty for all students,. $\endgroup$ – Brendan W. Sullivan Nov 27 '20 at 1:01
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    $\begingroup$ @alephzero I disagree. I find those two words very clear, once one familiarises themselves with their meanings after a period of getting used to it. "Injective" is certainly (imo) a better term to use than "one-to-one", for example, since the latter term confuses many students who may think this means "single-valued". $\endgroup$ – YiFan Nov 29 '20 at 9:34
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I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty.

The following three statements are logically equivalent to each other, but students will almost all find the first to be easier to understand than the second, and will not be mature enough to really understand the third (although they may like it for its brevity). However, mathematicians almost universally prefer the second definition.

  1. A function $f:X \to Y$ is not injective if and only if there exist $x_1, x_2 \in X$ with $x_1 \neq x_2$ and $f(x_1) = f(x_2)$.
  2. A function $f: X \to Y$ is injective if and only if $f(x_1)=f(x_2)$ implies $x_1=x_2$ for all $x_1,x_2 \in X$.
  3. A function $f: X \to Y$ is injective if and only if each element in the image of $f$ has a unique preimage.

It is easy to show a function is not injective: you just find two distinct inputs with the same output. Students can look at a graph or arrow diagram and do this easily. If given a function they will look for two distinct inputs with the same output, and if they fail to find any, they will declare that the function is injective. This conception fits extremely well with the first definition.

The second definition is equivalent, but this equivalence is not at all obvious to students. They are not accustomed to performing logical transformations as complex as this "on the fly". However, mathematicians almost universally prefer this definition (and for good reason: it leads to a much simpler proof structure when you actually want to prove that a function is injective, and it is much easier to use when you know a function is injective.)

Here is the symbolic proof of equivalence:

$$ \begin{align*} &\neg [ \exists x_1 \in X \exists x_2 \in X (x_1 \neq x_2) \wedge (f(x_1) = f(x_2))] \\ &\equiv \forall x_1 \forall x_2 \neg [(x_1 \neq x_2) \wedge (f(x_1) = f(x_2))]\\ & \equiv \forall x_1 \forall x_2 (x_1 = x_2) \vee \neg (f(x_1) = f(x_2))\\ & \equiv \forall x_1 \forall x_2 (f(x_1) = f(x_2)) \implies (x_1 = x_2) \end{align*} $$

I think students could have conceptual trouble with any line of this reasoning (even subconsciously). For instance, they could have trouble negating universally quantified statements, they could have trouble with de Morgan's laws, although I expect that most of them will get hung up on the last step (realizing that $p \implies q$ is equivalent to $\neg p \vee q$).

The second definition is hard for students to swallow for a number of reasons: we state that $f(x_1) = f(x_2)$ in the hypothesis. For many students, if we have given a different name to two variables, it is because the values are not equal to each other. Then in the conclusion, we say that they are equal! Then the student might say "why did I give them two different names in the first place!?!". We did so to imagine a world where $x_1$ and $x_2$ are not equal, and yet their function values agree. We want to rule that world out, which is why we have the implication. Students are used to only thinking about the $T \implies T$ part of the implication truth table, but here we are using that $T \implies F$ is false: we are ruling out the hypothesis being true while the conclusion is false by declaring that the implication is true.

The third definition adds an extra complication. We might say something like 3. as a clarifying remark. Since it is mostly devoid of logical complication (these are swept under the rug with the words "image", "preimage", and "unique"), this feels easier for the student. However, they are in danger because they do not actually have a rigorous understanding of the meaning of these words. This is the definition I think they are trying to mimic when they say "a function is one-one if every element in the domain has a unique image". Some students may also make this error because of a faulty linguistic assumption: thinking at "one to one" means "one element goes to one element". I love @Thierry's suggestion in another answer to replace "one to one" with "two to two" to address this linguistic trap.

This is my diagnosis of the problem, but I do not have great solutions for you. I see it in all of my upper level math classes: discrete math (where they are first introduced to injectivity), linear algebra, number theory, abstract algebra, etc. I do not have a good solution. Occasionally I will meet with a student intensively in office hours and discuss these points for hours. Some of these encounters do seem to fix the difficulty, but I cannot pinpoint what made it "click" for the student. It seems like an extremely persistent problem, and is probably indicative of some deep barriers to advancing in mathematics which I do not currently understand or know how to help people overcome.

EDIT:

One thing which has helped a bit (I think) is giving explicit names for the following distinct concepts:

A relation $R \subset X \times Y$ is said to be

  • single valued iff $(x,y_1) \in R$ and $(x,y_2) \in R \implies (y_1 = y_2)$.
  • total iff for all $x \in X$ there exist $y \in Y$ with $(x,y) \in R$.
  • injective iff $(x_1,y) \in R$ and $(x_2,y) \in R$ implies $x_1 = x_2$
  • surjective iff for all $y \in Y$ there exist $x \in X$ such that $(x,y) \in R$.

This way I can at least verbally discuss student errors: when students give the false definition you mention I can say "No, you are just saying that $f$ is single valued, which is part of the definition of a function".

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    $\begingroup$ The symmetry of the properties of relations is interesting: A relation is injective iff the transpose of the relation is single valued, and is surjective iff the transpose of the relation is total. However, students rarely confuse totality and surjectivity. This is also somewhat of a mystery to me. $\endgroup$ – Steven Gubkin Nov 26 '20 at 17:13
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    $\begingroup$ Does a textbook I wrote count? :-) $\endgroup$ – Brendan W. Sullivan Nov 27 '20 at 2:18
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    $\begingroup$ I looked more carefully: in fact, my definition is more like cdefintion 1.5" on your list: f is injective iff for all distinct x1, x2, it follows that f(x1), f(x2) are distinct. It combines the "distinct" part of definition 1 with the "if..then.." part of definiton 2, but uses the contrapositive of that conditonal. I hope that still counts! $\endgroup$ – Brendan W. Sullivan Nov 27 '20 at 2:26
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    $\begingroup$ (+1) This is a nice comprehensive list, and while I know you know and probably the OP and most others here know, but on the off-chance that someone things I don't know, you're not suggesting throwing all this at students. Instead, this is what you want to have in your toolbox in order to have different ways of presenting the idea as needed. Although I've only taught 6 (U.S. level) advanced undergraduate or graduate courses (that I can think of now), my experience has been that students all know what the concept is. What they don't always have a good command of (continued) $\endgroup$ – Dave L Renfro Nov 27 '20 at 8:41
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    $\begingroup$ is how to formally apply the property or to prove the property in various settings, and this applies to more than "injective", which is why I'm using "the property". The different mathematical formalisms of the property provide different avenues for letting the formalism guide you towards what you want. I thought I was going to say something a bit more specific and less flowery, but it seems to have gotten away from me, so I'll stop now! $\endgroup$ – Dave L Renfro Nov 27 '20 at 8:52
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On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say "a function is one-one if every element in the domain has a unique image".

Have you asked your students what they mean by "unique" when they say that? The reason I'm asking is that, by some commonly used definitions of the word, this statement is perfectly correct!

In particular, "unique" in English can have either of the following meanings (among others):

  1. single, without alternative, only one (as in "the equation has a unique solution"), or

  2. distinct from all others of its kind, not the same as any other (as in "ammonia has a unique smell" or "each genuine dollar bill has a unique serial number").

Indeed, I would even go so far as to suggest that, outside certain branches of mathematics, the second meaning is probably more common. And if that's the meaning your students are using, then they do have the right mental concept, even if they're expressing it in an unfortunately ambiguous way.

(Of course, the different meanings also have a lot of overlap in practice — for example, most serial and ID numbers tend to be unique in both senses of the word.)

Your students may not realize this potential ambiguity or, even if they do, may not be able to come up with a good alternative way of saying what they mean. Next time you encounter this phrasing, you might want to try simply prompting your students with a question like:

"When you say that 'every element has a unique image', do you mean that no element may have two different images, or that two different elements may not have the same image?"

That should at least give you a clearer idea of what your students are thinking, and whether their problem lies in understanding the concept or merely in explaining it. And, at the same time, you're also subtly pointing out the ambiguity and suggesting alternative ways of phrasing similar statements in the future.

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    $\begingroup$ Thanks that is a valuable suggestion. This had not crossed my mind. Next time I will ask the question you suggested. When they said this definition, I had asked them "How would you define a function?". And, then they said "uh-oh". So, I am not sure if they understood or not. The question you suggested will make it clear. $\endgroup$ – Divakaran Divakaran Nov 27 '20 at 11:53
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    $\begingroup$ @DivakaranDivakaran: If students are saying "uh-oh" in response to "How would you define a function?" then IMHO that's a deeper problem than what you have presented in the question. $\endgroup$ – Kevin Nov 27 '20 at 21:38
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    $\begingroup$ @DivakaranDivakaran: Your students apparently don't know what a function is, if they cannot give a definition of it. $\endgroup$ – Kevin Nov 28 '20 at 2:16
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    $\begingroup$ @Kevin Sorry, that is not what I meant. As soon as I ask for a definition of a function, they realise that every function would be one-one if one uses their definition. They can give a definition of a function. Perhaps, I didn't express myself clearly. $\endgroup$ – Divakaran Divakaran Nov 28 '20 at 7:38
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    $\begingroup$ @DivakaranDivakaran This excellent answer (and Ben Crowell's) led me to dig up and continue an old thread re: the ambiguity of 'unique'. This ambiguity is why, when describing injections using Defn 3 in Steven Gubkin's answer, I always modify it to "each element of the codomain has $\require{cancel}\cancel{\text{a unique}}$ at most one preimage" or "no two elements of the domain have the same image". $\endgroup$ – Ryan G Nov 29 '20 at 11:17
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The only example you give of what's going wrong is students' definition using the words "a function is one-one if every element in the domain has a unique image." I suppose it literally translates into a statement that for every x in the domain, there exists a unique y such that f(x)=y. This is really more like a description of the fact that in our standard way of defining functions, functions are always single-valued (although some people do still describe things like the inverse tangent as multiple-valued functions).

If a student offered this definition, and I was concerned that it represented a conceptual misunderstanding, I would probe more to ask what they meant. For example, you could ask them whether the sine is one-to-one by their definition.

If they respond with a correct conceptual explanation, but you're concerned about their ability to express themselves precisely in standard mathematical language, then that would be a separate issue. You could explain why the word "unique" in their phrasing is misleading. This would be a concern for math majors at the university level. If they're not going to be mathematicians, this seems to me like an issue of mathematical writing style that is not worth worrying about.

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    $\begingroup$ These are mathematics major students at the university. Given a function, they can determine correctly whether it is one-to-one or not. So, I am concerned about their ability to express precisely. I also explained why the word "unique" is misleading. They understand/accept the difference while I explain, but they tend to repeat the same mistake. $\endgroup$ – Divakaran Divakaran Nov 26 '20 at 14:36
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    $\begingroup$ @DivakaranDivakaran When you told them the word “unique” is misleading, did you give them another word to say instead? Because you can’t tell something to not say something without having something to replace it. I think the word they are looking for is “distinct”. A perfectly good definition of injective is “the elements of the domain all have distinct images” or alternatively, “distinct elements of the domain have distinct images”. $\endgroup$ – DavidButlerUofA Nov 27 '20 at 6:26
  • $\begingroup$ I told them you may say "distinct elements in the domain have distinct images". However, the next time, they stated the definition of a one-to-one function as "every element in the domain has a distinct image". They caught on to the word distinct but had not understood the idea. Some of the difficulty might be because my students are not native speakers of English. $\endgroup$ – Divakaran Divakaran Nov 27 '20 at 6:43
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In his Abstract Algebra book, John Fraleigh mentions that this is a common mistake beginning students make. Exercise 37 in Section 0 (7th edition) asks the reader to make a pedagogical case for using the terminology "two-to-two" instead of one-to-one. He doesn't expect the new terminology to actually take hold, of course, but just discussing it might make things clearer to students.

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    $\begingroup$ Ha! Two to Two is great! A much more fitting name. $\endgroup$ – Steven Gubkin Nov 26 '20 at 16:56
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    $\begingroup$ This approach is interesting, and I can see it might be helpful. I like it. A disadvantage is that "two-to-two" makes it less clear that an end-goal of defining an "injective function" is to provide the primary necessary condition for a function to have an inverse. So either we do the "hard" conceptual work first to understand the definition from the one-to-one approach and then slide into the notion of an inverse function, or we define injective from the two-to-two approach, deferring the conceptual work related to how it relates to inverse functions. But still, this is a refreshing idea! $\endgroup$ – user52817 Nov 26 '20 at 19:23
  • $\begingroup$ @user52817 Perhaps we could phrase rephrase "single valued" as "two from two". That is, if two distinct values are in the image of a function, they must come from two distinct inputs. Then we can see if a function $f$ is "Two-to-two" it's inverse relation must be "two-from-two", and hence a function on the image $f$. Typing out this comment, I am thinking this might be a bad idea simply because of the constant repetition of the word "two" (or to), which could create a hypnotizing effect. $\endgroup$ – Steven Gubkin Nov 27 '20 at 4:00
  • $\begingroup$ Thanks, that is indeed such a fitting name. $\endgroup$ – Divakaran Divakaran Nov 27 '20 at 6:24
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    $\begingroup$ I remember actually doing that exercise in some version of Fraleigh's book as a student, and I proposed the phrase "one from one" as an alternative. As in: "Pick an output. Where did it come from? Only one input." $\endgroup$ – Nick C Dec 2 '20 at 15:22
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You mention that these students are fresh out of high school. I'm not an educator but I can tell a little about how I first experienced this. What is hard in the beginning is to translate dense math language to an intuitive picture in your mind. You learn how to do this after some practice but in the beginning this can be daunting. You can guide your students in this proces by providing drawings and intuition and motivate the definition of a surjection.

Examples of this could be

  • Explain the word origin. Injection comes from "throw/hurl in/onto". We ofcourse know injection as putting something in something else. Surjection is defined as “a throwing over or on; an exaggeration, a hyperbole”. Does this make sense for the definition?
  • draw pictures (which you probably already do)
  • give concrete examples and counterexamples and explain why they are injective/not injective

Once you have solid intuition about injective functions the definition should become a natural statement. Like you mentioned these students don't have a solid background in proofs/logic so they may understand $f(a)=f(b)\implies a=b$ on a loose level but fail to see what this implies in practice.

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    $\begingroup$ I draw pictures and give a lot of examples and counter examples. I didn’t explain the word origin. Will keep that in mind. Thanks. $\endgroup$ – Divakaran Divakaran Nov 28 '20 at 16:47
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I find the view of solving equations very helpful, when I explain injective and surjective functions:

A function $f:X\to Y$ is

  • injective if the equation $f(x)=y$ has always at most one solution,
  • surjective if the equation $f(x)=y$ has always at least one solution, and
  • bijective if the equation $f(x)=y$ has always exactly one solution.
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  • $\begingroup$ I wonder if some will construe solve to mean in "closed form" (a la functions-are-formulae mentality). $\endgroup$ – Vandermonde Dec 1 '20 at 23:58

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