ECE608, Fall 2016, Quiz 2
Last Name: ___________________ First Name: ____________________
I certify that I have neither given nor received unauthorized aid on this quiz.
Signed: ___________________
Use only the space provided on this page to answer the following question(s).
Do not write your answers on the other side of the page.
For every one of the following claims, mark whether it is true or false. Prove your answer
only in the two cases where a proof is requested.
1.
3n = Θ(32n )
True/False
False
2.
3 = O(3 )
True/False
True
3.
3 = Ω(3 )
True/False
False
4.
32n = Θ(3n )
True/False
False
n
n
2n
2n
5.
3
= O(3 )
True/False
False
6.
32n = Ω(3n )
True/False
True
2n
n
Prove your answer
Prove your answer
Proof of 1: Assume by contradiction that there exist c1 > 0, c2 > 0 and n0 > 0 such that
0 ≤ c132n ≤ 3n ≤ c232n for all n ≥ n0 . Let us consider the inequality c132n ≤ 3n . This
implies c13n ≤ 1, or 3n ≤ 1/c1 , or n ≤ log3(1/c1). This cannot be satisfied together with
n ≥ n0 . Contradiction.
Proof of 6: We need to find constants c and n0 such that 32n ≥ c3n for all n ≥ n0 . From
32n ≥ c3n we obtain 3n ≥ c. With n0 = 1, this requires c ≤ 3. Possible constants are n0 = 1
and c = 3.
ECE608, Fall 2016, Quiz 2
Last Name: ___________________ First Name: ____________________
I certify that I have neither given nor received unauthorized aid on this quiz.
Signed: ___________________
Use only the space provided on this page to answer the following question(s).
Do not write your answers on the other side of the page.
For every one of the following claims, mark whether it is true or false. Prove your answer
only in the two cases where a proof is requested.
1.
2.
42n = Θ(4n )
True/False
False
2n
= O(4 )
True/False
False
2n
4
n
3.
4
= Ω(4 )
True/False
True
Prove your answer
4.
4n = Θ(42n )
True/False
False
Prove your answer
5.
4 = O(4 )
True/False
True
6.
4n = Ω(42n )
True/False
False
n
n
2n
Proof of 3: We need to find constants c and n0 such that 42n ≥ c4n for all n ≥ n0 . From
42n ≥ c4n we obtain 4n ≥ c. With n0 = 1, this requires c ≤ 4. Possible constants are
n0 = 1 and c = 4.
Proof of 4: Assume by contradiction that there exist c1 > 0, c2 > 0 and n0 > 0 such that
0 ≤ c1 42n ≤ 4n ≤ c2 42n for all n ≥ n0 . Let us consider the inequality c1 42n ≤ 4n . This
implies c1 4n ≤ 1, or 4n ≤ 1/c1 , or n ≤ log4(1/c1). This cannot be satisfied together with
n ≥ n0 . Contradiction.