HOMEWORK 1
SOLUTIONS
(1) Let
A
=
{
3
,
4
,
5
}
, B
=
{
3
,
4
}
, C
=
{
4
}
. Find
D
=
A
4
B
4
C
.
Solution:
Recall from class that
A
4
B
= (
A
\
B
)
∪
(
B
\
A
). That is,
A
4
B
contains all elements that lie in
A
but not in
B
and all elements that lie in
B
but
not in
A
. Note that we also have
A
4
B
= (
A
∪
B
)
\
(
A
∩
B
).
Thus
(
A
4
B
)
4
C
=
h
(
A
4
B
)
\
C
i
∪
h
C
\
(
A
4
B
)
i
=
h
(
A
\
B
)
∪
(
B
\
A
)
\
C
i
∪
h
C
\
(
A
∪
B
)
\
(
A
∩
B
)
i
=
A
\
(
B
∪
C
)
∪
B
\
(
A
∪
C
)
∪
C
\
(
A
∪
B
)
∪
C
∩
A
∩
B .
On the other hand
A
4
(
B
4
C
)
=
h
A
\
(
B
4
C
)
i
∪
h
(
B
4
C
)
\
A
i
=
h
A
\
(
B
∪
C
)
\
(
B
∩
C
)
i
∪
h
(
B
\
C
)
∪
(
C
\
B
)
\
A
i
=
A
\
(
B
∪
C
)
∪
A
∩
B
∩
C
∪
B
\
(
C
∪
A
)
∪
C
\
(
B
∪
A
)
.
A close inspection shows that the last term in both of the above equations are the
same.
(2) Suppose 70% of Californians like cheese, 80% like apples and 10% like neither.
What percentage of Californians like both cheese and apples?
Solution:
First let’s ask a different question. How many Californians like apples
or like cheese? If we add the percentages together we get 70%+80% = 150%. This
is clearly an overcounting. The problem is that we have counted those who like
both apples and cheese twice.
Since 10% of Californians like neither apples nor cheese, we know that 90% either
like apples or like cheese (including those who like both).
Hence, the
difference
between the percentages, 150%

90% = 60% answers the original question.
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 Spring '11
 ADEBOYE
 Graph Theory, Mathematical Induction, Vertex, Natural number, Berlin UBahn

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