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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '10
 any
 Graph Theory, Sets, Prime number, DI

Click to edit the document details