Here "is less" is interpreted as "eventually less for all values" and "plus a set" is interpreted as "plus any function of that set".
I never liked this notation for asymptotics and I always preferred the $f(x) \in O(g(x))$ style, but it's just notation in the end.
You can do this sort of "particular solution plus kernel" analysis on any linear operator, which gives one strategy for solving linear differential equations. e.g. (aD^2+bD+cI) is a linear operator (weighted sums and compositions of linear operators are linear), so you can potentially do that analysis to solve problems like af''+bf'+cf=g. In that context you say the general solution is to add a homogeneous solution (af''+bf'+cf=0) to a particular solution (my intro differential equations class covered this but didn't have linear algebra as a prereq so naturally at the time it was just magic, like everything else presented in intro diffeq).
A coset, quotients r + I, affine subspaces v + W, etc. Not literal sets but comparing some representative with a class label, and the `=, +` is defined not just on the actual objects but on some other structure used to make some comparison too.
I have always thought that expressing it like that instead of f(x) ∈ O(g(x)) is very confusing. I understand the desire to apply arithmetic notation of summation to represent the factors, but "concluding" this notation with equality, when it's not an equality... Is grounds for confusion.
it's a notation for "some element of that set"
f(x) = g(x) + O(1)
f(x) - g(x) = O(1)
I think first we should teach "f in O(g)" notation, then teach the above, then observe that a special case of the above is the "abuse of notation" f(x) = O(g(x)).
You get:
f(x) - g(x) ≤ O(1)
f(x) - g(x) = O(1)
f(x) - g(x) ≤ O(1)
Programmers wringing their hands over the meaning of f(x)=O(g(x)) never seem to have manipulated any expression more complex than f(x)=O(g(x)).