I have a follow-up to a previous question: True or false or not-defined statements.
(In the following I might use the word statement incorrectly.)
In that question/answer I learned that the "statement" that $\frac{1}{0} =1$ is not true or false because the expression isn't a well-formed formula (i.e. the left have side is not well defined). I do understand that one might make the "symbol" $\frac{1}{0}$ in certain areas of mathematics. For this question, though, I am just interested in the math we teach in say an undergraduate calculus class.
My follow-up question is what about the statement
$\frac{1}{x} = 1$ for all real numbers $x$
Here the statement makes fine sense if we had said "for all values of $x\neq 0$".
But would this statement also be not-true and not-false because the expression is not defined/well-formed since $x=0$ is part of the statement?
(Even though I am interested in how one might answer this in a calculus class, I would like the strict logic answer. I know that we often say and write things that technically/strictly speaking not true, but when say teaching about true and false statements, I would like to keep things as precise as possible. Even though we can sometimes overlook some things in an undergraduate class, I also think it is wrong to teach stuff that is outright wrong.)