Usage of "Let" in math Is the usage of "let" in the following parts correct with respect to usage and language?
(1): Let $x \in \mathcal{R}$ . I read that let must always be coupled with "be" (ams.org/publications/authors/AMS-StyleGuide-online.pdf).
(2): Let $x_{t}^{P}$ denote the position of Person P at time t.
 A: The first is fine, although it might be read aloud as "Let $x$ be in ${\cal R}$." You could argue that it should more properly be written as something like, "Let $x$ be an element of ${\cal R}$," but it's hard to justify the extra verbiage for something that's already completely clear. The symbol $\in$ denotes set inclusion, not the English phrase "is an element of." If you don't like omitting the verb "to be," (or, effectively using the inclusion symbol $\in$ as a verb rather than an adjective) replace it with something like, "Take $x\in {\cal R}$," or "Fix $x\in {\cal R}$."
The second isn't quite the right usage. If you mean that $x_t$ is the position at time $t$, just write something like, "Define $x_t$ to be the position at time $t$," or, "Let $x_t$ be the position at time $t$." The reader or the author is the one doing the defining, not the number $x_t$. As written, the sentence sounds like the position is just some deterministic function of the time $t$, rather than $x_t$ itself.
