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Exercises on Functions and Induction. Due:
Tuesday, November 10th (at the beginning of
the class)
Reminder: the work you submit must be your own. Any collabora
tion and consulting outside resourses must be explicitely mentioned
on your submission.
Please, use a pen. 30 points will be taken oﬀ for pencil written
work.
1. Find the domain and range of the following function
f
. It assigns to
each string of bits (that is, of 0s and 1s) the number equal to twice of the
number of zeros in that string. For example
f
(011001010101010) = 16.
Solution
By deﬁnition
f
assigns a value to “each string of bits” thus the domain
is the set of all bit strings.
It’s easy to see that the range is the set of all positive even numbers.
An odd number can not be a value of
f
since the value is twice the
number of zeros. Any even number 2
k
has a preimage: a string of
k
zeros.
2. Determine whether or not the function
f
:
Z
×
Z
→
Z
is onto, if
f
((
m,n
)) =
m

n
.
Solution
Yes it is onto. For any number
k
we have
f
((
k,
0)) =
k

0 =
k
.
3. Give an explicit formula for a function from the set of all integers to
the set of positive integers that is onto but is not onetoone.
Solution
The formula is
f
(
x
) =

x

+ 1.
That is
f
(
x
) is the absolute value of
x
plus 1. The deﬁnition is correct
because for any integer except for zero

x

is a positive integer and

x

+ 1 is a positive integer. For 0 we shall have
f
(0) = 1.
1
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 Spring '09
 jcliu
 Math

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