O(1)
< O(log(n))
< O(sqrt(n))
< O(n)
< O(n log(n))
< O(n²)
< O(2ⁿ)
< O(n!)

@0foldcv I don't remember ever seeing this combination of < and O. Is it supposed to mean something different than = O(...)?

@11011110 @0foldcv strict subset

@axiom @11011110 Exactly. I used "<" in the sense of "strict subset" (maybe I should have used "⊂").

Saying something like
O(log(n)) = O(n) seems too much of an abuse of notation for my taste (because O(n) = O(log(n)) is false).

Not sure if "=" is always an abuse of notation for Landau symbols, but at least it's convenient for things like f(x) = g(x) + O(x²). In the previous case, I would prefer it meant equivalence.

@0foldcv @axiom My interpretation is that you should think of "= O" as being one piece of notation, to be read as meaning something like "grows at most as quickly as". Because it makes sense that way when it doesn't make sense as a separate equality relation and function-to-set-of-functions operator. That's why I react so negatively when I see things like "< O". You're neither using the standard notation nor using a combination of operations that makes any sense.

@11011110 @0foldcv just never use $$= O$$. I always mentally translate it to $$\in O$$

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