# Asymptotic Notations - Big Omega

Big-Omega is the lower bound of a function is denoted as notation (Ω).

Definition: Suppose we have two functions f(n) and g(n), then the function f(n) = Ω(g(n) if and only if there exists two positive constants c and k, such that f(n) >= c * g(n) for all n >= k

For example:

Suppose, if f(n) = 2 * n + 3, then

2 * n + 3 >= 1 * n for all n >= 0.

In the above equation,
c = 1, k = 0, and g(n) = n

Now we know that, 1< log(n) < n^(1/2) < n < n log(n) < n^2 < n^3 < — < 2^n < 3^n <---- n^n.
In the above comparison, 1< log(n) < n^(1/2) are lower bound.

This symbolizes,

1. f(n) = Ω(n) — True
2. f(n) = Ω(n^2) — False
3. f(n) = Ω(1) — True, and so on…

**Finally, in Big-Omega, we take the nearest g(n), which in this case is Ω(n).

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What is the difference between a knife and an orange? Yup, two different things. The connection there is that, among its other uses, you can use a knife to peel an orange.

The big omega notation is mathematical notation used to talk about asymptotic growth of functions. Saying that “𝑔

is Omega of 𝑓" (or writing “𝑔∈Ω(𝑓)”) means that 𝑔 grows asymptotically at least as quickly as 𝑓

.

Big omega notation is useful to express lower bounds. The sentence “each comparison-based sort runs in Ω(𝑛log𝑛)

" means that the (worst-case) time complexity of any comparison-based sort is a function that grows at least as quickly as the function 𝑛log𝑛

.

Among other things, you can also use this notation to talk about the best-case performance of algorithms. For example, you can say things like “on any input of size 𝑛

, MergeSort will perform Ω(𝑛log𝑛) steps".