COT 3100 Section 2 Homework #2 Fall 2000 Solutions
1)
a)
{0,1,4,16}
b)
{1,4,16}
c)
{12,6,0}
2)
a)
2
7
. This is the exact same question as asking how many subsets of the set
{2,4,6,8,10,12,14} there are, since you are forced to pick all the odd numbers
every time.
b)
8
C
3
*2
7
. Since you are choosing the odd numbers and the even numbers
independently to form a cartesian product, you multiply. You can choose 3 odd
numbers in
8
C
3
ways. (In part a we determined the total number of ways of
picking the even numbers.)
c)
8
C
3
*
7
C
5
. Same principle as question b, but this time you must pick exactly 5 even
numbers which can be done in
7
C
5
ways.
d)
2
14
. This is just like counting the total number of subsets of the set {2,3,4,.
..,15}.
e)
10. The subsets are
∅
,{1},{2},{3},{4},{5},{1,2},{1,3},{1,4}, and {2,3}.
f)
2
13
– 1. Since you can’t have 7 or 14 in the sets, you basically have to choose non
empty subsets from the set {1,2,3,4,5,6,8,9,10,11,12,13,15}. There are 2
13
subsets
of that set and we must subtract out the empty set.
If we interpret the question
differently, such that we count the set {2,3,7}, for example, because it doesn’t
have 14, then our answer is (2
15
– 1) – 2
13
. To see this, basically, the only non
empty subsets we don’t want to count are those with BOTH 7 and 14 in them. So,
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 Fall '09
 Set Theory, Empty set, Natural number

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