Big O vs Theta Θ vs Big Omega Ω Notations

1. Big O notation (O): 

It defines an upper bound on order of growth of time taken by an algorithm or code with input size. Mathematically, if f(n) describes the running time of an algorithm; f(n) is O(g(n)) if there exist positive constant C and n0 such that,

0 <= f(n) <= Cg(n) for all n >= n0

n = used to give upper bound a function. 
If a function is O(n), it is automatically O(n-square) as well. 

2. Big Omega notation (Ω) : It defines a lower bound on order of growth of time taken by an algorithm or code with input size. Let f(n) define running time of an algorithm;
f(n) is said to be Ω(g (n)) if there exists positive constant C and (n0) such that 

0 <= Cg(n) <= f(n) for all n >= n0

n = used to given lower bound on a function 
If a function is Ω(n-square) it is automatically Ω(n) as well. 

3. Theta notation (Θ) : 

It defines exact order of growth of time taken by an algorithm or code with input size. Let f(n) define running time of an algorithm. f(n) is said to be Θ(g(n)) if f(n) is O(g(n)) and f(n) is Ω(g(n)).

Mathematically, 

0 <= f(n) <= C1g(n) for n >= n0
0 <= C2g(n) <= f(n) for n >= n0

Merging both the equation, we get :  

0 <= C2g(n) <= f(n) <= C1g(n) for n >= n0

The equation simply means there exist positive constants C1 and C2 such that f(n) is sandwich between C2 g(n) and C1g(n). 

Difference Between Big oh, Big Omega and Big Theta : 

For more details, please refer: Design and Analysis of Algorithms.