anseladdams

That's true. You're right.

> If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?

Couldn't there be cases where we don't have a known tight asymptotic bound but do have an upper and/or lower bound? And although it's an abuse of notation, you do often see big-O used in place of Theta. From CLRS:

"In the literature, we sometimes find O-notation informally describing asymptotically tight bounds, that is, what we have defined using Theta-notation."

No, this is flat out wrong. Big O is an upper bound on an asymptotic growth rate. Big Omega is a lower bound on the asymptotic growth rate. Big Theta is a tight bound on the asymptotic growth rate. These are independent to the average case, best case, or worst case run time of a given algorithm.

Quicksort has an average run time of Theta(n lg n). Equivalently, its average run time is O(n lg n) and Omega(n lg n). It has a worst case run time of Theta(n^2). Equivalently, its worst case run time is O(n^2) and Omega(n^2).

> Actually we don't use it informally as big-Theta,

Wrong again.