The definition of $\bigcirc$ used by mathematicians is not convenient for engineers.
So in practice we ignore that definition and use our own.
The mathematical definition is, roughly, that if $f(n)=\bigcirc\big(g(n)\big)$ then $f(n)$ grows no faster than $g(n)$, that there exists some value K such that for values of $n$ of interest and larger than of interest $f(n)\le Kg(n)$
Which is kind of stupid for engineers, because by that definition an algorithm that takes time $\bigcirc(n)$ also takes time $\bigcirc(n^2)$, $\bigcirc(n!)$, etcetera.
So, Knuth defined $\large\Omega$, which means, roughly, that there exists some value K such that for values of $n$ of interest and larger than of interest $f(n)\ge Kg(n)$
Which is also stupid for the same reason.
So what all engineers do in practice is use $\bigcirc$ to mean that the mathematical definition of $\bigcirc$ is true, *and* Knuths definition of $\large\Omega$ is also largely true, so when we say that an operation take that much time, we mean that it takes no more than that much time, *and frequently takes something like that much time*.
So, by the engineer's definition of $\bigcirc$, if an algorithm takes $\bigcirc(n)$ time it does *not* take $\bigcirc(n^2)$ time.
