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What does big omega and big theta terms stands for? What are their purposes?

Big-oh concerns with the "less than or equal to" relation between functions for large values of the variable. It is also possible to consider the "greater than or equal to" relation and "equal to" relation in a similar way. Big-Omega is for the former and big-theta is for the latter.

Definition (big-omega): Let f and g be functions from the set of integers (or the set of real numbers) to the set of real numbers. Then f(x) is said to be Ω(g(x)) , which is read as f(x) is bigomega of g(x) , if there are constants C and n0 such that

| f(x) | ≥C | g(x) |

whenever x > n0 .

Definition (big-theta): Let f and g be functions from the set of integers (or the set of real numbers) to the set of real numbers. Then f(x) is said to be θ( g(x) ) , which is read as f(x) is big-theta of g(x) , if f(x) is O( g(x) ), and Ω( g(x) ) . We also say that f(x) is of order g(x) .

For example, 3x2 - 3x - 5 is Ω( x2 ) , because 3x2 - 3x - 5 ≥x2 for integers x > 2 (C = 1 , n0 = 2 ) .

Hence by Theorem 1 it is θ( x2) .

In general, we have the following theorem:

Theorem 4: an xn + ... + a1 x + a0 is θ( xn ) for any real numbers an , ..., a0 and any non-negative number n .