Learning Goal: I’m working on a discrete math case study and need a sample draft to help me learn.

Problem 14.20

Identify each of the following states as true or false. Proof that your response is correct.

(a) n2 ∼ n2 + n

(b) 3n = O (2n)

(c) n = Θ ( 3n3

n2−1 )

*Required Problem Response
Replace this text with your response to this problem.*

Problem 14.34

False Claim.

2n = O(1). (1)

False Proof. The proof is by induction on n where the induction hypothesis P(n) is the assertion

(1).

Base Case: P(0) holds trivially.

Inductive Step: We may assume P(n), so there is a constant c > 0 such that 2n ≤ c ⋅ 1.

Therefore

2n+1 = 2 ⋅ 2n ≤ (2c) ⋅ 1,

which implies that 2n+1 = O(1). That is, P(n + 1) holds, which completes the proof of the

inductive step.

We conclude by induction that 2n = O(1) for all n. That is, the exponential function is bounded by

a constant. QED.

Identify a major mistake in the above False Proof.

Origenal question inside the pdf.

- Explore the content in Sections 14.1 – 14.7 of the textbook
*Mathematics for Computer Science*. Focus on the content related to the learning objectives listed on the overview page for this module. https://ccsf-math-115.github.io/textbook/mcs_2018_cropped.pdf#view=FitH&page=638

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