Homework 7 Solutions
Kyle Chapman
November 29, 2011
137.2
This question eﬀectvely only asks how many ways are there to choose three sets of size 6
,
4
,
2. Thus the
answer is
(
12
6
,
4
,
2
)
= 13860.
137.8
There are two ways to calculate this. The ﬁrst is by multiplying the odds of each number being one that
was picked, which gives the result of
10
50
9
49
8
48
7
47
6
46
5
45
=
.
0000132. Alternatively, we can get the result by
comparing the number of ways the numbers could be picked, namely
(
50
6
)
to the number of ways that the
desired numbers can be picked, which gives
(
10
6
)
. Taking the ratio of these two gives the same result.
147.2
1
We can get this rather directly from the DeMorgan’s Laws.
A
∪
(
B
∩
C
) =
{
x

x
∈
Aorx
∈
(
B
∩
C
)
}
=
{
x

x
∈
Aor
(
x
∈
Bandx
∈
C
)
}
=
{
x

(
x
∈
Aorx
∈
B
)
and
(
x
∈
Aorx
∈
C
)
}
=
{
x

x
∈
(
A
∪
B
)
andx
∈
(
A
∪
C
)
}
= (
A
∪
B
)
∩
(
A
∪
C
)
2
Here we can get a simple counter example from
A
=
B
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 Fall '08
 Bigelow
 Sets, Elementary arithmetic, Exponentiation, Equivalence relation, Empty product

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