The Grammar According to West

You encounter abusive language/notation basically everywhere in math. Open up a calculus/real analysis textbook. A lot of the old ones write sequences in the curly brace/set {x_n} notation:

"let {x_n} be a sequence"

As the author points out, a sequence is a function. The statement {x_n} is the set of terms of the sequence, its range. A function and its range are two different things. And also sets have no ordering. It might seem like a minor thing, but I thought we were trying to be precise?

A second example: at the high school level, I'm pretty sure a lot of textbooks don't carefully distinguish between a function and the formula defining the function very well.

The author of this web page has a section on what he calls "double duty definitions". Personally, I don't find anything wrong with the language "let G=(V,E) be a graph". G is the graph and we're simultaneously defining/naming its structure. So, some of this is a matter of taste. And, to some extent, you just have to get used to the way mathematicians write.

> abusive language/notation basically everywhere in math

In most cases it is not as much abusing notation as overloading it. If you think of the context of a formula (say, adjacent paragraphs) as its implicit arguments (think lambda captures in c++), then it is natural that curly braces can denote both a set and a sequence, depending on this implicit input.

Such context dependent use of symbols is actually rather convenient with a little practice.