review1solns.pdf - CMPS 101 Midterm 1 Review Solutions to...

CMPS 101
Midterm 1 Review
Solutions to selected problems
2. State whether the following assertions are true or false. If any statements are false, give a related statement
which is true.
a. ?(?) = ?(?(?)) implies ?(?) = ?(?(?))
Related True Statement: ?(?) = ?(?(?)) implies ?(?) = ?(?(?))
b. ?(?) = ?(?(?)) if and only if ?(?) = Ω (?(?))
c. ?(?) = 𝛩(?(?)) if and only if lim
𝑛→∞
(?(?)/?(?)) = 𝐿 , where 0 < 𝐿 < ∞ . False
3. Prove that Θ(?(?)) ⋅ Θ(?(?)) = Θ(?(?) ⋅ ?(?))
.
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4. Let ?(?) and ?(?) be asymptotically positive functions (i.e. ?(?) > 0 and ?(?) > 0 for all sufficiently
large n ), and suppose that ?(?) = Θ(?(?)) . Does it necessarily follow that
?(𝑛)
= Θ (
1
?(𝑛)
) ? Either prove
this statement, or give a counter-example.
1
such
1
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