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Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009.
Sets, Functions, Cardinality
1. Determine whether each of the following are injective, surjective, both (bijective), or neither. Prove your answer
(a)
f
:
N
→
N
f
(
x
) =
x
2
Injective
(b)
f
:
R
→
R
f
(
x
) = 3
x
+ 4
Bijective
(c)
f
: (0
,
1)
→
(1
,
∞
)
f
(
x
) =
1
x
Biective
2. For a function
f
:
A
→
B
, given
S
⊆
A
, let
f
(
S
) :=
{
y
∈
B
:
∃
x
∈
S,f
(
x
) =
y
}
.
Prove that if
f
is onetoone, then
f
(
S
∩
T
) =
f
(
S
)
∩
f
(
T
) for any
S,T
⊆
A
.
We have to show that the two
SETS
f
(
S
∩
T
) and
f
(
S
)
∩
f
(
T
) are equal. So we have two steps:
1) We show
f
(
S
∩
T
)
⊂
f
(
S
)
∩
f
(
T
):
Let
b
∈
f
(
S
∩
T
).
Then this means that there exists an
x
∈
S
∩
T
such that
b
=
f
(
x
).
Thus
x
∈
S
and
x
∈
T
.
So
f
(
x
)
∈
f
(
S
) and
f
(
x
)
∈
f
(
T
)
Therefore
f
(
x
)
∈
f
(
T
)
∩
f
(
S
).
But
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 Summer '09
 RobBayer
 Math, Sets

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