- Let
*f*and*g*be a functions from the non-negative integers into the positive real numbers - For some real constant
*c*> 0 and some non-negative integer constant*n*_{0}- O(
*g*) is the set of functions*f*, such that:*f*(*n*) ≤ c **g*(*n*) for all*n*≥*n*_{0}

- Ω(
*g*) is the set of functions*f*, such that:*f*(*n*) ≥ c **g*(*n*) for all*n*≥*n*_{0}

- Θ(
*g*) = O(*g*) ∩ Ω(*g*)- Θ(
*g*) is the asymptotic order of g or the*order*of g

- Θ(

- O(

*f*(*n*) ∈ O(*g*(*n*)) means that there are positive constants *c* and *n*_{0} such that *f*(*n*) ≤ *c***g*(*n*) for all values *n* ≥ *n*_{0}

- Is
*n*∈ O(*n*^{2})?- Yes,
*c*= 1,*n*_{0}= 2 works fine

- Yes,
- Is 10
*n*∈ O(*n*)?- Yes,
*c*= 11,*n*_{0}= 2 works fine

- Yes,
- Is
*n*^{2}∈ O(*n*)?- No; no matter what values for
*c*and*n*_{0}we pick,*n*^{2}>*c***n*for big enough*n*

- No; no matter what values for

which of these are true?

- For
positive integers*all**m*,*f*(*m*) <*g*(*m*). - For
positive integer*some**m*,*f*(*m*) <*g*(*m*). - For
positive integer*some**m*_{0}, andintegers*all positive**m*>*m*_{0},*f*(*m*) <*g*(*m*). - 1 and 2
- 2 and 3
- 1 and 3

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