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.

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.

You get:

  f(x) - g(x) ≤ O(1)

  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)).

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)).