Context: This is for introductory linear algebra course, near the beginning.

As a sort of "exit survey" after one of my lectures, I would like to ask my students to try and define what "unique" is from a mathematical perspective. I would then talk about that next time. However, I am not sure

  1. whether that's a good idea; or
  2. how to phrase the question so that I don't get answers like "it means you're really special".
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    $\begingroup$ @Namaste I want this to be more of a "pre-assessment" prior to discussing unique solutions $\endgroup$ – Avi Steiner Sep 12 '19 at 21:03
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    $\begingroup$ This question greatly confuses me, since uni- means "one". Ergo, a function with a unique solution is (no pun intended) one with one solution. $\endgroup$ – RonJohn Sep 13 '19 at 6:02
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    $\begingroup$ To follow up: merriam-webster.com/dictionary/unique "being the only one". It's my experience that mathematicians and scientists use existing words with defined meanings. $\endgroup$ – RonJohn Sep 13 '19 at 6:04
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    $\begingroup$ @XanderHenderson definitely not colloquial and vernacular English. That's where twaddle like "it means you're really special" comes from. Latin, Greek and traditional English definitions are what to look at. $\endgroup$ – RonJohn Sep 13 '19 at 18:10
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    $\begingroup$ @Namaste you're also using modern definitions instead of their roots. The root of "rational" is ratio. Transcendental numbers transcend ("be or go beyond the range or limits of what can be defined with a ratio") the real, and the term "right angle" derives from the Latin for "upright angle". $\endgroup$ – RonJohn Sep 14 '19 at 20:19

To avoid misinterpretations, I'd give it to them in a mathematical sentence and ask them to explain what they believe 'unique' means in this context.

For lack of knowing where you are in the course, I might give them an exit ticket such as

Consider the sentence, "The equation $x^3 - 2x + 3 = 0$ has a unique solution." What do you think the word "unique" means in this context? Contrast this with the ways you might use "unique" in standard English.

However, this sentence has a problem in that the word 'unique' can meaningfully only be interpreted in one way (or perhaps I'm uncreative); perhaps give them a mathematical sentence which is not quite so obvious.

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    $\begingroup$ Context matters. $\endgroup$ – paul garrett Sep 12 '19 at 22:07
  • $\begingroup$ Why would mathematical usage be any different from normal language usage? The dictionary meaning is quite clear -- exactly one of a kind, no other such item. I'm fully (and sorrowfully) aware that many people misuse the word, e.g., "this is a very unique..." $\endgroup$ – Carl Witthoft Sep 13 '19 at 11:31
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    $\begingroup$ @CarlWitthoft Although the use of 'unique' in this way is related to 'singular' which has the same literal meaning, but has been used to mean "special" rather than necessarily "unique" since at least the 19th century. $\endgroup$ – Cubic Sep 13 '19 at 13:12
  • $\begingroup$ @Cubic and that usage is still wrong $\endgroup$ – Carl Witthoft Sep 13 '19 at 15:52
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    $\begingroup$ @OpalE Honestly, I wouldn't be surprised if the word "colloquial" confuses more students than it helps, as students may not know what it means. Perhaps "Contrast this with the ways that people use 'unique' when they speak or write English in an informal or nonmathematical setting," or "Contrast this with the ways that you use 'unique' in non-mathematical contexts," or something similar. $\endgroup$ – Xander Henderson Sep 13 '19 at 18:33

I think that a linear algebra course is the perfect venue to have students develop their understanding of the mathematical concept of "uniqueness." A formative assignment such as the one you suggest is a good, initial step.

Later on, as the course progresses, students might even come to appreciate the mathematical phrase unique up to unique isomorphism.

Here is an example in linear algebra. A vector space of dimension $n$ over a field $k$ is unique, because any two such objects are isomorphic by a linear transformation. However, they are not unique up to unique isomorphism. If instead, you work in the category of finite dimensional vectors spaces with an ordered basis over a fixed field $k$, then two vector spaces of the same dimension are unique up to unique isomorphism.


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