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Chapter 3
73
(b)
1
1
can be defined on the range of I,that is, R ' {2}.
(c) Let
y
=
1
1
(x), x
=I
2~
Then
x
=
f(y)
=
~.
This implies
y(2x
 4) =
x.
Since
x
=I
2,
2y
1
x
'
x
we have
y
=
2x
_ 4' Thus
1
1
(x)
=
2x
_ 4;
7. (a) The values are 7,5,3,1,2,4,6.
(b) Let
n
E
N. We must show that
n
=
I(a)
for some integer
a.
If
n
is even, let
a
=
~n.
If
n
is odd,
let
a
=
~(n

1).
'
8. A function
I
on a set
A
is a reflexive relation if and only if
I
=
i
is the identity on
A.
Certainly,
the identity function is reflexive. Conversely, if
I
is reflexive, then for any
a
E
A,
(a, a)
E
f.
Since
(a, x), (a, y)
E
I
imply
x
=
y,
it follows ,that for any
a
E
A,
(a, x)
E
I
if and only if
x
=
a.
Thus
I=~
,
9. Lots of functions R + R are symmetric, for example, the function defined by
I(x)
=
x.
10. The greatest integer function is one example. For any
a, b,
e E R, if
b
=
La
J
and e =
L
b
J, then e =
b
because
b
is an integer, so e =
La
J
.
11. (a) This is false:
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 Graph Theory

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