> computer science students should be familiar with the standard f(x)=O(g(x)) notation
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.
I think the confusion is because strictly speaking $f(x) = O(g(x))$ is an abuse of notation. $O(g(n)), \Theta(g(n))$ and friends are sets. We can't say that a function equals a set, or that a function "is less" than another function, but notoriously mathematics runs on javascript, so we try to do something instead of giving a type error.
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.
To me it seems similar to the + C on an antiderivative (or more generally, quotient objects). Technically, you are dealing with an equivalence class of functions, so a set. But it's usually counterproductive to think of it that way (and when you study this stuff properly, one of the first things you do is prove that you (usually) don't need to, and can instead use an arbitrary representative as a stand-in for the set), so you write F(x)+C.
I feel its not that bad an abuse of notation as kinda consistent with other areas of mathematics -
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.
Although, when I learned foundations of mathematics, every function was a set, and if you wanted them, you'd get plenty of junk theorems from that fact.
> computer science students should be familiar with the standard f(x)=O(g(x)) notation
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.
Given this possible confusion, is it still valid to say the following two expressions are equivalent as the article does?
f(x) = g(x) + O(1)
f(x) - g(x) = O(1)
I think the confusion is because strictly speaking $f(x) = O(g(x))$ is an abuse of notation. $O(g(n)), \Theta(g(n))$ and friends are sets. We can't say that a function equals a set, or that a function "is less" than another function, but notoriously mathematics runs on javascript, so we try to do something instead of giving a type error.
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.
To me it seems similar to the + C on an antiderivative (or more generally, quotient objects). Technically, you are dealing with an equivalence class of functions, so a set. But it's usually counterproductive to think of it that way (and when you study this stuff properly, one of the first things you do is prove that you (usually) don't need to, and can instead use an arbitrary representative as a stand-in for the set), so you write F(x)+C.
I feel its not that bad an abuse of notation as kinda consistent with other areas of mathematics -
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.
Although, when I learned foundations of mathematics, every function was a set, and if you wanted them, you'd get plenty of junk theorems from that fact.
"Everything is an object" is for boys, "everything is the empty set composed with copies of itself via the axiom of pairing" is for men ;)
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