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mid1review

# mid1review - CMPS 101 Midterm 1 Review Problems 1 Let f(n...

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CMPS 101 Midterm 1 Review Problems 1. Let ) ( n f and ) ( n g be asymptotically non-negative functions which are defined on the positive integers. a. State the definition of )) ( ( ) ( n g O n f = . b. State the definition of )) ( ( ) ( n g n f ω = 2. State whether the following assertions are true or false. If any statements are false, give a related statement which is true. a. )) ( ( ) ( n g O n f = implies )) ( ( ) ( n g o n f = . b. )) ( ( ) ( n g O n f = if and only if )) ( ( ) ( n f n g Ω = . c. )) ( ( ) ( n g n f Θ = if and only if L n g n f n = )) ( / ) ( ( lim , where < < L 0 . 3. Prove that )) ( ) ( ( )) ( ( )) ( ( n g n f n g n f Θ = Θ Θ . In other words, if )) ( ( ) ( 1 n f n h Θ = and )) ( ( ) ( 2 n g n h Θ = , then )) ( ) ( ( ) ( ) ( 2 1 n g n f n h n h Θ = . 4. Let ) ( n f and ) ( n g be asymptotically positive functions (i.e. 0 ) ( > n f and 0 ) ( > n g for all sufficiently large n ), and suppose that )) ( ( ) ( n g n f Θ = . Does it necessarily follow that Θ = ) ( 1 ) ( 1 n g n f ? Either prove this statement, or give a counter-example. 5. Let ) ( n g be an asymptotically non-negative function. Prove that = Ω )) ( ( )) ( (( n g n g o .

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