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54
Solutions to Exercises
7. (a) Jumps in the graph o.f
y
=
L
2x
 3
J
o.ccur whenever
2x
 3 is an integer, that is, at
{~
I
n
E
Z}.
(b) Jumps in the graph o.f
y
=
rix
+
7J occur whenever
ix
+ 7 is an integer, that is, at
{4n
I
n
E
Z}.
(c) Jumps in the graph o.f
y
=
L
~
J
occur whenever
~(x
+
3) is an integer, that is, at
{x
I
x
=
5n3,n
E
Z}.
8. (a) [BB] The answer is ''yes.''· By definition o.f "flo.o.r", we kno.w that
kx
is the unique integer satisfy
ing
n
 1
<
kx
<
n.
Thus
!! 
1 .(
x
<
!!. Since !! 
1
<
!! 
1 we have
!! 
1
<
x
<
!!.

k
k

k· k

k
k' k

k
Thus
x
=
LIJ.
.
(b) The multiples o.f 3 in the indicated interval are 3,6,9,
...
,3k,
where
3k
=
L
50000 J. Thus
k
=
L
50~OO
J
= 16666.
9. [BB] We kno.w that Lx J is an integer less than o.r equal to.
x
and that Ly J is an integer less than o.r equal
to.
y.
Thus LxJ
+
LyJ is an integer less than o.r equal to.
x
+
y.
Since Lx
+
yJ is the greatest such
integer, we get the desired result.
10. We kno.w that
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 Summer '10
 any
 Graph Theory

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