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 lim𝑛→∞(?(?)/?(?)) = 𝐿, where 0 < 𝐿 <∞. False
3.Prove that Θ(?(?)) ⋅ Θ(?(?)) = Θ(?(?) ⋅ ?(?))
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4.Let ?(?)and ?(?)be asymptotically positive functions (i.e. ?(?) > 0and ?(?) > 0for 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.
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