CS173 Discrete Mathematical Structures
Fall 2006
Homework #5
due Sunday, October 1, 2006, 8:00 a.m.
1. Given functions f: B → C and g: A → B, if f and (f ◦ g) are one-to-one, does it follow
that g is one-to-one? Prove your response.
2. Suppose f is a bijection from A to B. Suppose g is a bijection from B to C.
Prove that (g ◦ f) is a bijection from A to C.
3. Let
xxf 10)(
3.1. Find f ◦ (2f)
3.2. What is the domain and the co-domain of f ◦ (2f)?
4. True or false?
4.1.
)3(2
2nn
nO
4.2.
))(log(2
100
nnO
n
4.3.
))!((2 nnO
n
4.4.
)()log(
001.1
nOnn
4.5.
)()log(
0001.0
nnOnn
4.6.
)(!
10
10
nn
4.7.
)2(1
101
O
5. Give a tight big-O estimate on the following functions:
5.1.
))!log(log(
)log(10
)(
10
n
nn
nf
5.2.
n
n
nf
3
1
...
3
1
3
1
3
1
2
1
...
2
1
2
1
2
1
)(
210
210
6. Prove or disprove:
6.1. If f1(x) = O(g(x)) and f2(x) = O(g(x)), then g = (f1(x)+ f2(x))
6.2.
If f1(x) = O(g(x)) and f2(x) = O(g(x)), then (f1◦f2)(x) = O(g(x))
6.3. If f(x) = O(g(x)) then
)2(2
)()( xgxf
O
6.4 If f(x) = x2+x+10 then f(x) = O(x2)