168
Solutions to Exercises
(b) The number of elements which belong to
Al
and
A2
but to no other
Ai
is
IA
1
nA2
,(A3UA4)1.
We
determine this number and then multiply by 6 because there are six ways for elements to belong
to exactly two of the sets. Since
(AI
n
A2)
n
(A3
U
A4)
=
n
n
A3)
U
n
n
A4),
I(A
1
n
A2)
n
U
A4)1
=
IAI
n
n
A31
+
IAI
n
n
A41IAl
n
n
A3
n
n
n
A41
=
5
+
5
1
=
9.
Thus
I(A
1
n
A2)
,
U
A4)1
=
IAI
n
A21
I(AI
n
A2)
n
U
A4)1
=
12 
9
=
3. There
are 6 x 3
=
18 elements in exactly two of the four subsets.
11. (a) [BB] Let
A
and
B
be the set of integers between 1 and 500 which are divisible by 3 and 5,
respectively. The question asks for
IA
UBI.
This number is
IAI
+
IBI 
n
BI
=
l5~0
J
+
l5g0
J 
l5
1
0
5
0
J
=
166 + 100 
33
=
233.
(b) [BB] Let
A
and
B
be as in (a) and
G
be the set of integers between 1 and 500 which are divisible
by 6. The question asks for
A, (BUG)I
and so the number we want is
IAIIAn
(BuG)I.
Now
An (B
U
G)
=
(A
n
B)
U
n
G),
so the number of elements here is l5
1
0
5
0
J
+
l5~0
J
l5
3
0
0
0
J
=
33 + 83  16
=
100. Finally, we obtain
,
(B
U
G)I
=
166 
100
=
66.
12. (a) Since 500
=
225
3
,
an integer is relatively prime to 500 if and only if it is not divisible by 2 or by
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory, Sets

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