# The Big O and big Theta definitions

1. Using the definition of big-O, show that 2n element of O(6*n^2)

2. Using the definition of big-O, show that 6n^2 not element of O(2n)

3. Show that 6n^2 element of THETA(n^2)

Do not use limits, just the big O and big Theta definitions.

Please show the definitions.

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#### Solution Preview

Big-O definition: We say f(n) = O(g(n)), if we can find a sufficiently large constant M>0, such that for sufficiently large n, we have f(n) <= M*g(n).

1. Let M = 1. For all n>=1, we have 6n^2 - 2n = n(6n - 2) >= 1*(6*1 - 2) = 4 > 0. Thus 2n <= ...

#### Solution Summary

Big O and big Theta definitions are illustrated.

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