The term "unique" for functions and operations

Question

This is long so...

TLDR: Proposing the math community steer away from the misleading term unique, with respect to functions and algebraic operations. Instead, use unambiguous. Why not? Analysis below.

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I just started to get back into math after a year off, and after cracking open Pinter's book on intro abstract algebra I was reminded of a phrasing issue I've always had. The phrase "uniquely defined" or "uniquely mapped". For example, the operation $a*b$ defined by normal multiplication $ab$ on $\mathbb{R}$ is considered uniquely defined—the operation takes any ordered pair in $\mathbb{R}$ and produces one, and only one value in $\mathbb{R}$. As Pinter describes uniquely defined:

In other words, the value of $a*b$ must be given unambiguously.

After a few moments, it becomes clear to the experienced math reader that this is a defining property shared with functions—an operation can't have more than one output. We get it.

But as I was shaking off the cobwebs, I remembered this terminology being a small barrier for me when I first started learning about functions and operations (and other objects), one that, I'm ashamed to admit, slowed me down once again.

Reflecting over the past hour, I've begun to view my learning friction as the result of poor encoding—the the word unique doesn't align with intuition. When I first read the word unique I interpret it as well... the normal word unique. As Merriam-Webster defines it.

being the only one

Applying that to function/operation context, there's a few interpretations to be had, but one the first and most natural one that comes to my mind is that each ordered pair is assigned a value unique to that ordered pair. Which is stronger definition of uniqueness. Clearly, the example above, with normal multiplication, does not meet this stronger definition, since $(a,b)$ and $(b,a)$ map to the same value under normal multiplication on $\mathbb{R}$. In other words, $(4, 3)$ is not the "only one" since it shares a value with $(2,6)$, $(3, 4)$, etc.

I acknowledge that in this function/operation context there is an argument that mappings in their entirety are always unique, since we're using ordered values—one unique ordered pair, for example, maps to a value. But this is not how uniqueness is portrayed or described in textbooks, in textbooks emphasis is on function output or operation value being singular. And I further find this justification for the term unique to be weak, since it leans on the uniqueness of ordered values and not the combining process of the elements.

Of course, mathematical terms and definitions are not meant to align perfectly with normal language. So I'm not arguing that we restrict math terminology to only language aligned perfectly with intuition, however we do (and should) always make an effort to name things appropriately. I think that here, the term unique violates naming principles and is misleading. I propose that Pinter's own terminology better. Let's call this defining property of functions and operations unambiguousness. E.g., functions and operations are unambiguous. Unambiguity seems like the word we actually mean, which is probably why Pinter wrote it.

I wrote this because, anything that can be done to remove learning friction should be.

So... My Question

Can you see a good reason to not shift gears and teach functions/operations as unambiguous. Any obvious faults? And of course, I'd be telling students about classic terminology.

Answer 1

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way.

Students have trouble with the notion of a function because it's hard. The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions about it.

Reflecting over the past hour, I've begun to view my learning friction as the result of poor encoding—the the word unique doesn't align with intuition.

I think you should be skeptical of this reflection. When we're learning, there's a common tendency to credit the very last thing that makes the idea fall into place, and miss that all the seemingly aimless struggling that came before it was actually what prepared us for that last insight.

Maybe if functions had been described to you using "unambiguous" rather than "unique", you would have understood them sooner - but I think it's more likely that you would have needed about the same amount of thinking about the idea to sort through all the possible misunderstandings, and some other aspect would appear to have been the last insight you needed.

Answer 2

In Germany, we already do this. A function is introduced as an unambiguous mapping in 7th grade (~13 years). While I don't have any data on this, I doubt that German students do significantly better due to this choice of words.

Answer 3

I personally use the following terminology:

A relation $R \subset A \times B$ is said to be single-valued if $(a,b_1) \in R$ and $(a,b_2) \in R$ implies $b_1 = b_2$.

A relation $R \subset A \times B$ is said to be total if for all $a \in A$ there exists $b$ in $B$ with $(a,b) \in R$.

A relation which is both single valued and total is a function.