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Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class....

Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class.

a)n^15log n + n^9 is O(n^9 log n)

b) 15^7n^5 + 5n^4  + 8000000n^2 + n is Θ(n^3)

c) n^n is Ω (n!)

d) 0.01n^9  + 800000n^7 is O(n^9)

e)   n^14 + 0.0000001n^5 is Ω(n^13)

f)    n! is O(3n)

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Answer #1

a) There does not exist any positive constant c and n0 such that

0 <= n^15log n + n^9 <= c*n^9 log n

Hence, this statement is false.

b) There does not exist any constant c which always guarantees that

0 <= 15^7n^5 + 5n^4  + 8000000n^2 + n <= c * n3

Hence, this statement is false.

c) Clearly,

1 * 2 * .... * n < n * n * ..... * n

Hence, n! < n^n

This means that n^n is Ω (n!)

NOTE: As per Chegg policy I am allowed to answer specific number of questions (including sub-parts) on a single post. Kindly post the remaining questions separately and I will try to answer them. Sorry for the inconvenience caused.

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