MATH 74 HOMEWORK 5: DUE MONDAY 3/5
1.
For each of the following functions, tell me:
•
Is the function injective?
•
Is the function surjective?
You do not need to prove your answers, or do anything besides answer
yes or no for each question.
(The point is just to get you thinking
about these concepts. If you want to give full proofs, feel free.)
(a)
f
:
N
→
N
given by
f
(
x
) =
x
2
(b)
f
: [0
,
∞
)
→
R
given by
f
(
x
) =
x
3
(c)
f
: [

π/
2
, π/
2]
→
[

1
,
1] given by
f
(
x
) = sin
x
(d)
f
:
P
(
N
)
→ P
(
N
) given by
f
(
A
) =
A
∪ {
1
,
2
,
3
}
(e)
f
:
P
(
N
)
→ P
(
N
) given by
f
(
A
) =
{
x
∈
N
:
x
∈
A
}
2.
Let
A
be a finite set.
Let
B
denote the set of all functions with
domain
A
and codomain
{
0
,
1
}
. Prove that #(
P
(
A
)) = #(
B
).
[Hint: come up with a function from
P
(
A
)
→
B
, or vice versa, and
prove that it is a bijection.]
3.
Definition.
If
f
:
X
→
Y
is a function, and
S
⊂
X
, we let
f

S
denote
the function whose domain is
S
, whose codomain is
Y
, and whose rule
is the same rule as
f
. (In other words,
f

S
is the function
S
→
Y
you
get by thinking of
f
as a function only on
S
and not on all of
X
.) In
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 Fall '07
 COURTNEY
 Math, Set Theory, Empty set, Finite set, Basic concepts in set theory, Bijection

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