CS173 Discrete Mathematical Structures
Fall 2006
Homework #4
due Sunday, September 24, 2006, 8:00 a.m.
1. Let Ai = {i, i+1, i+2,…}. Let Bi be the set of all nonempty bit strings of length not
exceeding i. (For example, B2 = {0, 1, 00, 01, 10, 11}). Find:
a.
n
i
i
A
1
b.
n
i
i
A
1
c.
n
i
i
B
1
d.
n
i
i
B
1
2. Which of the following are functions from the domain to the codomain given?
Which of these functions are injective? Which are surjective? Which are
bijective?
a. f : Z → R,
9
1
)(
2
x
xf
b. f: Z → R,
9)(
2
xxf
c. f: N → R,
9)(
2
xxf
d. f: N → N,
2
)( x
xf
e. f: N → N,
otherwise 1
even is if 1
)( x
xx
xf
3. How many elements are in the union of four sets if each of the sets has 100
elements, each pair of sets shares 50 elements, each three of the sets share 25
elements, and there are 5 elements in all four sets?
4. Let f be a function from the set A to the set B. Let Q be a subset of B. We define
the inverse image of Q to be the subset of A containing all pre-images of all
elements of Q. We denote the inverse image of S by f -1(Q), so that
f -1(Q) = {a A | f (a) Q}.
Let S and T be subsets of B. Show that:
a. f -1(S T) = f -1(S) f -1(T)
b. f -1(S T) = f -1(S) f -1(T)