Can we avoid confusion over using "let" as a quantifier?
I've encountered the following misunderstanding. I pose a question (to undergraduates in the U.S.), for example:
Let $P$ be a polygon of $n$ vertices. Is it true that every triangulation of $P$ has the same number of triangles?
This question depends on what constitutes a "triangulation," but assume the student know that. The answer is Yes: every triangulation of $P$ consists of $n-2$ triangles.
Here is the problem I encounter. The students apparently don't understand that "Let $P$ be a polygon" means, let your mind run over all possible polygons, so $P$ is an "arbitrary" polygon, in that it can be anything that fits the definition of a polygon (which the students also know). They wonder, well, maybe $P$ is a convex polygon, and should I answer specific to convex polygons?
This example doesn't quite illustrate the problem because the answer is always Yes. But when the answer is sometimes Yes, sometimes No, they seem to get confused over the quantifier. I think it may come down to the meaning of the phrase: "Let $A$ be a $B$." Let $p$ be a point in the plane $\mathbb{R}^2$—meaning any point in the plane, an "arbitrary" point in the plane. Let $P$ be a polygon, meaning any polygon. Let $T$ be a triangulation of a set of $n$ points. Does every triangulation of $n$ points have the same number of triangles? (No, not always.)
Have you encountered this confusion in your teaching? If so, how do you circumvent it?
1 Answer
To clarify the scope of a mathematical statement, replace the word let with more explicit universal quantifiers when introducing a variable. Instead of saying let p be a point, try framing the prompt as for any point p in the plane or consider an arbitrary polygon p. Using terms like for all or for every signals that the property must hold regardless of the specific instance chosen, which helps students distinguish between a single, fixed example and a general case. Another effective strategy is to explicitly state the boundaries of the domain. If you want students to consider all polygons, clarify this by saying let p be any polygon, regardless of its convexity. By adding these short qualifiers, you remove the ambiguity that leads students to wonder if they should be looking for a special case. It shifts their focus from guessing the intended subset to evaluating the truth of the statement across the entire category. You can also encourage students to test the boundaries of the definition themselves. When presenting a problem, ask them to mentally construct a few different versions of the object in question. If they ask whether the polygon must be convex, treat it as a productive moment to discuss whether the property depends on that specific trait. This teaches them to view the variable as a placeholder for the entire class of objects rather than a specific, hidden example. Finally, distinguish between existential and universal claims by using concrete examples during your lectures. When you introduce a variable, briefly contrast what changes if the object is fixed versus what happens when it is arbitrary. This helps students internalize that the word let is a tool for defining the scope of their investigation. By consistently using precise language, you provide a model for how they should interpret and formulate their own mathematical arguments.
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