Big-O notation $=$ or $\equiv$

Big-O notation is used to show the relative growth rates of functions. For example, $2x^2+3x=O(x^2)$ or $4x^2-2x+5=O(x^2)$, implying for $x\to\infty$, the limit of their ratio is a non-zero constant.

On this post and Wikipedia page, it was stated several issues with the use of the symbol $=$.

Can the symbol of equivalence $\equiv$ used instead to resolve the issues. Then it can be written: $2x^2+3x\equiv 4x^2-2x+5\equiv O(x^2)$ just like in modular arithmetic. It would imply something like the first function is congruent to the second function mod $C$. Consequently it would be transitive.

Yes, there can be an inconsistency with the fact that $O(f(x))$ represents a class of functions. Belonging of a function to a class of functions is represented with $\in$ just like in $f(x)\in C^2$ (the class of twice differentiable functions). Does it make sense or is it a serious violation of the math notations?

• It is an acceptable notation, but non-standard. Therefore, you should define what you mean by it whenever you use it. But it's fine, in principle. It's better to use standard notation though, so you should stick to $=$ or $\in$. – AccidentalFourierTransform May 27 '18 at 15:08
• What is just like in modular arithmetic? What is the modulus? It is common to write $2x^2 +3x \asymp 4 x^2 -2 x +5$. – quid May 27 '18 at 15:11
• @quid That symbol (whose TeX shortcut I have forgotten) is often used for logarithmic equivalence in my field, I wouldn't consider it common as a substitute for Big Oh in general. – Ian May 27 '18 at 16:18
• First of all, $2x^3+3x=O(x^2)$ does not mean that $\lim_{x \to \infty} (2x^2+3x)/x^2$ is a nonzero constant. It means that $\limsup_{x \to \infty} |2x^2+3x|/x^2$ is finite (possibly zero). Second, I've never seen anyone write $\equiv$ for this. If you want to be more precise (objecting to the use of $=$ in a non-referentially transparent and also non-symmetric manner) then you should use $\in$. – Ian May 27 '18 at 16:20
• @Ian It is not a substitute. $f \asymp g$ is used to express that $f$ is $O(g)$ and $g$ is $O(f)$, what is also written as Theta as you mentioned; in other words $|f/g|$ and $|g/f|$ are bounded (leaving aside subtleties when with 0 etc). This is somewhat common in analytic number theory. If the quotient tends to $1$ it would be $f \sim g$. But, yes, meaning of such symbols is not uniform over different fields. – quid May 27 '18 at 16:27

As long as you get used to "=" in big-oh and little-oh meaning set inclusion, you shouldn't have any problems. I still use "=" instead of the more accurate "$\in$".

Just remember that these are transitive relations (aRb and bRc implies aRc).

Note that, for positive functions, big-oh is reflexive (aRa) but not symmetric (aRb implies bRa) but little-oh is neither.

• Big Oh is not symmetric. Big Theta is symmetric. Confusingly, people often write Big Oh when they mean (and can prove) the relevant Big Theta relation. – Ian May 27 '18 at 16:22
• You are right - I will fix. Thanks. Upvoted. – marty cohen May 27 '18 at 16:33