CS 173: Discrete Structures, Fall 2010
Homework 5 Solutions
This homework contains 5 problems worth a total of 50 points. It is due on Friday, October
8th at 4pm.
1.
Functions [8 points]
For each of the following functions, state what its image is, whether the function is
onto, and whether it is onetoone. If the image is awkward to describe exactly, give
an approximate highlevel description rather than (say) copying the function definition
into setbuilder notation.
(a)
f
:
Z
→
Z
such that
f
(
k
) =
k
mod 7.
Solution:
The image of
f
is
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
. This function is neither oneto
one, nor onto.
(b)
g
:
R
2
→
R
such that
g
(
x,y
) =
⌊
x
⌋
+
y
.
Solution:
The image of
g
is
R
. This function is not onetoone, but it is onto.
(c)
h
: (
Z
+
)
2
→
R
such that
h
(
p,q
) = 2
p
+
1
q
, where
Z
+
is the positive integers.
Solution:
The image of
h
is a really wierd set. It consists of a little sequence of
numbers near each positive power of two. The offset (from the power of 2) starts
off with 1 and then continues with a set of increasingly small fractions. Because
all the values in the image are clustered near powers of two, the image doesn’t
cover anything like all of
R
, so it’s not onto.
The offsets are small enough that the numbers associated with one power of two
are separated from those associated with other powers of two, so this function is
onetoone.
(d)
k
:
Z
→
Z
such that
k
(
x
) = 2
x
2
+
x
.
Solution:
The image of
k
is
{
0
,
1
,
3
,
6
,
10
,
15
,
21
,...
}
. Notice that the difference
between the numbers in the list increases by one each time and the gaps eventually
become large. So there are many integers not included in this set. This function
is onetoone, but not onto.
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 Spring '08
 [email protected]
 Natural number, sK

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