The reason it's preferred to use "=" instead of "\in" is because the way that Landau notation is generally used in practice is as a kind of ellipsis or placeholder. For example, the Taylor expansion e^x = 1 + x + O(x^2). I could just as well write e^x = 1 + x + ..., but the former conveys more meaning about what is hidden behind the ellipsis. It's an abuse of notation, but in the contexts that it's used, it's not clear what additional clarity using "\in" would bring over "=". Maybe also that big O is mainly used as a notation to facilitate doing calculations, less describing what family a function belongs to. Here are Knuth's thoughts, which I agree with: https://micromath.wordpress.com/2008/04/14/donald-knuth-calc...

▲ marcosdumay 3 hours ago | parent | next [-]

You are talking about a completely different concept than the GP's.

That's why clear notation is important. Yours is kinda fine, but would be better with "≃".

	▲	Maxatar 2 hours ago \| parent [-]
		But it's not completely different and a "≃" would not mean the same thing, in fact it would weaken the statement. e^x ≃ 1 + x + O(x^2) would only assert that lim (x->0) (e^x)/(1+x) = 1. However "e^x = 1 + x + O(x^2)" means that for some function r(x) belonging to the set O(x^2), e^x is exactly equal to 1 + x + r(x). Another way to rewrite that equation that eliminates the "abuse of notation" would be: `e^x − (1 + x) ∈ O(x^2)` The particular r(x) in O(x^2) which makes it strictly equal is being left out, that's true, and usually it's left out for brevity or practical reasons or even because it's not even be known what r(x) is... but nevertheless it is not an asymptotic equation or an approximation, it is exactly equal to the value on the right hand side for some particular r(x) the exact details of which are being omitted for one reason or another.

▲ svat an hour ago | parent | prev [-]

See also Terence Tao's comments at https://terrytao.wordpress.com/2022/05/10/partially-specifie... which say things even more strongly (I had collected a link to it at https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...):

> The symbol ∈ only is a viable solution in a portion of the use cases. For instance, an assertion such as O(n)⋅O(n) = O(n²) would not be correctly describable as O(n)⋅O(n) ∈ O(n²). Perhaps O(n)⋅O(n) ⊂ O(n²) would be defensible, but now one has to devote a non-trivial amount of thought into deciding which of the connectives =, ∈, ∋, ⊂, ⊃ to use in a given context. For instance the assertion “Since sin(y) = sin(x) + O(|y−x|), we have sin(x+O(1/n)) = sin(x) + O(1/n)” would now become “Since sin(y) ∈ sin(x) + O(|y−x|), we have sin(x+O(1/n)) ⊂ \sin(x) + O(1/n)”. Using the equality sign for all of these use cases instead is more intuitive and corresponds more closely to how the verb “is” (“to be”) is actually used in mathematical English.

and

> … Nevertheless most of us still often think in mereological terms rather than set-theoretic or first-order terms […] without requiring translation to set theory or first order logic; indeed, such a translation would only serve to slow that mathematician down as he or she would usually have translate it back into mereological form in order to wield it effectively. Because of this, I think it is worth adjusting our notational conventions to more closely align with our actual thought processes… I don’t see much advantage in interpreting each instance of the O() notation in the exponential type bound f(n) = O(\exp(O(nᴼ⁽¹⁾))) or the calculation (1 + O(1/n))ᴼ⁽ⁿ⁾ = \exp(O(1/n)⋅O(n)) = \exp(O(1)) = O(1) (for n sufficiently large), in terms of ideals.