Big O Notation “is element of” or “is equal”

People are always having trouble with “big $O$” notation when it comes to how to write it down in a mathematically correct way.

Example: you have two functions $n\mapsto f(n) = n^3$ and $n\mapsto g(n) = n^2$

Obviously $f$ is asymptotically faster than $g$. Is it $f(n) = O (g(n))$ or is it $f(n) \in O(g(n))$?

My prof says that the first one is wrong but is a very common practice, therefore it is used very offten in books. Although the second one is the right one.

Why is that so?

Answer

I really like Wikipedia’s note on this:

The statement “$f(x)$ is $O(g(x))$” […] is usually written as $f(x) = O(g(x))$. Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, $O(x) = O(x^2)$ is true but $O(x^2) = O(x)$ is not. Knuth describes such statements as “one-way equalities”, since if the sides could be reversed, “we could deduce ridiculous things like $n = n^2$ from the identities $n = O(n^2)$ and $n^2 = O(n^2)$.”

For these reasons, it would be more precise to use set notation and write $f(x) \in O(g(x))$, thinking of $O(g(x))$ as the class of all functions $h(x)$ such that $|h(x)| \leq C|g(x)|$ for some constant $C$. However, the use of the equals sign is customary. Knuth pointed out that “mathematicians customarily use the $=$ sign as they use the word ‘is’ in English: Aristotle is a man, but a man isn’t necessarily Aristotle.”

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Source : Link , Question Author : Blnpwr , Answer Author : Community

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