MATH S104 Lecture 4 Inclass Problem Solutions
July 2, 2009
Problem 1
: Let
X
=
{
1
,
3
,
5
}
and
Y
=
{
a,b,c,d
}
. Draw an arrow diagram that defines a function
f
with each of the following characteristics, or explain why such a function cannot exist.
a)
f
is not a welldefined function.
a
b
c
1
3
5
d
a
b
c
1
3
5
d
The arrow diagram on the left is not welldefined because 3 has no image in
Y
. The
diagram on the right is not a welldefined function because 1 is mapped to multiple
images in
Y
.
b)
f
is not a welldefined function, but
f

1
is.
a
b
c
1
3
5
d
a
b
c
1
3
5
d
c)
f
is a function that is onetoone but not onto.
a
b
c
1
3
5
d
f
is not onto because b has no preimage in
X
.
d)
f
is a function and
f

1
is a function.
Such a function is not possible with the specified domain and codomain, because
Y
≠
X
.
1
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Problem 2
*
: Find at least three di
ff
erent sequences beginning with the terms 1,2,4 whose terms
are generated by a simple formula.
Some correct answers are:
•
a
n
=
2
n
,
n
=
0
,
1
,
2
,...
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 Spring '10
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 Mathematical Induction, Recursion, Inductive Reasoning, Mathematical logic, inductive step

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