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COT 3100
Summer 2001
Quiz #4 (Solutions)
06/28/2001
1. (4 pts) Let
A
= {
a
,
b
,
c
},
B
= {
a
,
b
}, and
f
= {(
a
,
b
), (
b
,
b
), (
c
,
a
)}. Then
f
:
A
→
B
.
What are
f
(
a
),
f
(
b
), and
f
(
c
)?
f
(
a
)=
b
,
f
(
b
)=
b
,
f
(
c
)=
a
.
2. (7pts) Let
f
: R
→
R be defined by the formula:
3
5
2
)
(
+
=
x
x
f
(R – the set of real numbers).
Prove that
f
is surjective and injective and find a formula for
To prove that
f
is injective we need to show that if
f
(
x
) =
f
(
y
), then
x
=
y
. So, assume
(2
x
+5)/3=(2
y
+5)/3. It implies that 2
x
+5=2
y
+5, and
x
=
y
( by trivial algebra).
To prove that
f
is surjective we should demonstrate, that for any
y
∈
R we can find
x
∈
R, such that
f
(
x
)=
y
. So, let
y
be any element of R and (2
x
+5)/3=
y
. Then by simple
algebra we get 2
x
+5 = 3
y
,
x
= (3
y

5)/2. We see, that for any
y
∈
R we can always find
x
= (3
y

5)/2 ,
x
∈
R, such that
f
(
x
) = [2
⋅
(3
y

5)/2+5]/3=[3
y

5 + 5]/3 =
y
.
f

1
(
y
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