[charcircuit]

>It also shows why Theta is better than both

Big Theta for some functions is an empty set. Take for example bubble sort. It's O(n^2) and Ω(n), but there is no function that asymptotically bounds it above and below. In fact it is better if there isn't one as it means there exists fast cases. For bubble sort it's fast cases are ones where data is almost sorted and requires to scans of the array. If you know this is true of the input data you know bubble sort is more appropriate than something like merge sort which is Ω(nlog(n)). Now the O(n^2) worst case can still be significant so you may want to consider an alternate algorithm that doesn't break down as bad.

As the other reply pointed out we are talking about functions that bound some other function (in this context the amount of steps / operations / time an algorithm will take) when looking at how it behaves asymptotically.

[tsimionescu]

Note that "the amount of steps / operations / time an algorithm will take" is not in general a mathematical function, as for the same input size, different inputs will give different numbers of steps. For bubble sort for example there doesn't exist a function f(n) = number of steps bubble sort will take for an input of size n. There does exist a function best_case_complexity(n) = number of steps bubble sort will take for the best input of size n; and a function worst_case_complexity(n) = number of steps bubble sort will take for the worst input of size n.

Then, we can say best_case_complexity(n) = Theta(n), and worst_case_complexity(n) = Theta(n^2). We can also define the set of all functions representing the number of steps bubble sort will take for input of a certain "shape" of size n. We can then indeed say that any function in this set will be <= worst_case_complexity(n) and >= best_case_complexity(n), so this set will indeed be a subset of Omega(n) and of O(n^2).