57 lines
2.5 KiB
XML
57 lines
2.5 KiB
XML
Section 8.5: 1, *5*, 9, *11*, *15*, *23*, 27
|
||
|
||
Section 8.6: 3, *5*, *7*, 13, *21*, *25*
|
||
|
||
= 8.5-5
|
||
Find the number of elements in $A_1 union A_2 union A_3$ if there
|
||
are 100 elements in each set and if
|
||
1. The sets are pairwise disjoint.
|
||
$100+100+100=300$
|
||
2. There are 50 common elements in each pair of sets and no elements in all three sets.
|
||
$100+100+100-50-50-50=150$
|
||
3. There are 50 common elements in each pair of sets and 25 elements in all three sets.
|
||
$100+100+100-50-50-50+25=175$
|
||
4. The sets are equal.
|
||
$100+100+100-100-100-100+100=100$
|
||
|
||
= 8.5-11
|
||
Find the number of positive integers not exceeding 1000 that are not divisible by $3,17, "or" 35$
|
||
|
||
$1000-333-58-28+19+9+1-17=610$
|
||
|
||
= 8.5-15
|
||
How many bit strings of length eight do not contain six consecutive 0s?
|
||
|
||
$2^8-12-4-1+7+2=248$
|
||
|
||
= 8.5-23
|
||
Write out the explicit formula given by the principle of inclusion–exclusion for the number of elements in the union of six sets when it is known that no three of these sets have a common intersection.
|
||
$ |A_1union A_2 union A_3 union A_4 union A_5 union A_6| =\
|
||
|A_1|+|A_2|+|A_3|+|A_4|+|A_5|+|A_6|-|A_1 inter A_2| - |A_1 inter A_3| - |A_1 inter A_4| - inter |A_1 inter a_5| - |A_1 inter A_6|\
|
||
- |A_2 inter A_3| - |A_2 inter A_4| - |A_2 inter A_5| - |A_2 inter A_6|\
|
||
- |A_3 inter A_4| - |A_3 inter A_5| - |A_3 inter A_6| - |A_4 inter A_5| - |A_4 inter A_6| - |A_5 inter A_6|
|
||
$
|
||
|
||
= 8.6-5
|
||
Find the number of primes less than 200 using the principle of inclusion–exclusion.
|
||
|
||
$A_n =$ Numbers divisible by `n`
|
||
$
|
||
"Number of primes" &= 200 - abs(A_2 union A_3 union A_5 union A_7 union A_11 union A_13) - 1 + 6 \
|
||
&= 205 - |A_2| - |A_3| - |A_5| - |A_7| - |A_11| - |A_13| \
|
||
&quad + |A_2 inter A_3| + |A_2 inter A_5| + ... + |A_11 inter A_13| \
|
||
&quad - |A_2 inter A_3 inter A_5| - ... \
|
||
&quad + |A_2 inter A_3 inter A_5 inter A_7| + ... \
|
||
&quad - |A_2 inter A_3 inter A_5 inter A_7 inter A_11| - ... \
|
||
&quad + |A_2 inter A_3 inter A_5 inter A_7 inter A_11 inter A_13| \
|
||
&= 205 - 100 - 66 - 40 - 28 - 18 - 15 \
|
||
&quad + 33 + 20 + 14 + 9 + 6 + 4 + 2 + 1 + 1 + 1 + 1 + 1 + 1 \
|
||
&quad - 6 - 4 - 2 - 2 - 1 - 1 - 1 - 1 - 1 - 1 \
|
||
&quad + 1 + 1 + 1 + 1 + 1 \
|
||
&quad - 0 - 0 - 0 - 0 \
|
||
&quad + 0 \
|
||
&= 46
|
||
$
|
||
= 8.6-7
|
||
How many positive integers less than 10,000 are not the second or higher power of an integer?
|
||
= 8.6-21 |