1. Section 16 Inclusion/Exclusion
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2/16/2019
MATH 106, Section 16

2. Set Notation A set is any well-defined collection of objects.
U represents the universal set (i.e., the universe).
A  B means that
A  B represents
A  B represents
~A represents
#A represents
A is a subset of B, that is, every item in A must also be an item in B. (Every set is a subset of U.)
the intersection of A and B, that is, the set of all items that are in both A and B.
the union of A and B, that is, the set of all items that are in at least one of A or B.
the complement of A, that is, the set of all items that are not in A.
the size of A, that is, the number of items that are in A.
2/16/2019
MATH 106, Section 16

3. #1
Consider a universal set consisting of the positive integers up to (and including) 12, that is, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. We define the subsets
A = multiples of 2
B = multiples of 3
C = multiples of 4
D = multiples of 5
Use set notation to display each subset; then find each of the following:
#A = #B =
#C = #D =
~A = ~B =
~C = ~D =
{2, 4, 6, 8, 10, 12}
{3, 6, 9, 12}
{4, 8, 12}
{5, 10}
2/16/2019
MATH 106, Section 16

4. Consider a universal set consisting of the positive integers up to (and including) 12, that is, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. We define the subsets
A = multiples of 2
B = multiples of 3
C = multiples of 4
D = multiples of 5
Use set notation to display each subset; then find each of the following:
#A = #B =
#C = #D =
~A = ~B =
~C = ~D =
{2, 4, 6, 8, 10, 12}
{3, 6, 9, 12}
{4, 8, 12}
{5, 10} 6 4 3 2
{1, 3, 5, 7, 9, 11}
{1, 2, 4, 5, 7. 8, 10, 11}
{1, 2, 3, 5, 6, 7, 9, 10, 11}
{1, 2, 3, 4, 6, 7, 8, 9, 11, 12}
2/16/2019
MATH 106, Section 16

5. AB = AC = BD = AB = AC = BD =
In each section of the Venn diagrams displayed, list the members of the set that section represents.
{6, 12}
{4, 8, 12}
{ } or 
{2, 3, 4, 6, 8, 9, 10, 12}
{2, 4, 6, 8, 10, 12}
{3, 5, 6, 9, 10, 12} 1 5 7 11 2 4 8 10 6 12 3 9 A B
2/16/2019
MATH 106, Section 16

6. 1 3 5 7 9 11 2 6 10 4 8 12 A C 1 2 4 7 8 11 3 6 9 12 5 10 B D
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MATH 106, Section 16

7. Complete each of the following formulas, and then verify that each is true:
#(AB) = #A + #B
#(AC) = #A + #C
#(BD) = #B + #D
– #(AB) 8 6 4 2
– #(AC) 6 6 3 3
– #(BD) 6 4 2
2/16/2019
MATH 106, Section 16

8. #2
Consider a universal set consisting of the positive integers up to (and including) 30, that is, U = {1, 2, …, 29, 30}. We define the subsets
A = multiples of 2
B = multiples of 3
C = multiples of 5
Use set notation to display each subset. In each section of the Venn diagram displayed, list the members of the set that section represents. Then find each of the following:
{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}
{3, 6, 9, 12, 15, 18, 21, 24, 27, 30}
{5, 10, 15, 20, 25, 30}
#A = #B =
#C = #(AB) =
#(AC) = #(BC) =
#(ABC) = 15 10 6
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MATH 106, Section 16

9. 1 7 11 13 17 19 23 29 2 4 8 14 6 12 18 24 3 9 21 27 16 22 26 28 30 A 10 20 15 B 5 25 C
2/16/2019
MATH 106, Section 16

10. Complete the following formula, and then verify that it is true:
#(ABC) =
#A + #B + #C
– #(AB)
– #(AC)
– #(BC)
+ #(ABC) 15 10 6 5 3 2 1
This is 22, which is what we count for ABC from the Venn diagram.
#A = #B =
#C = #(AB) =
#(AC) = #(BC) =
#(ABC) = 15 10 6 5 3 2 1
2/16/2019
MATH 106, Section 16

11. U = {1, 2, …, 29, 30}
A = multiples of 2 B = multiples of 3 C = multiples of 5
Note the transition from words to set notation and vice versa:
= multiples of each (all) of 2 and 3 and 5
= multiples of (at least) one of 2 or 3 or 5
= multiples of none of 2 and 3 and 5
= not a multiple of (at least) one of 2 or 3 or 5
ABC
ABC
~(ABC)
OR (~A)(~B)(~C)
~(ABC)
OR (~A)(~B)(~C)

12. For next class, do the following problems in the Section 16 Homework:
Problem #3: Use #U – #(AB) = #U – [#A + #B – #(AB)]
freshmen taking mathematics
all freshmen
freshmen taking computer science
Problem #4: Use #(ABC) =
#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)
children who play tennis
children who play soccer
children who play baseball
Problem #5: Use #U – #(AB) = #U – [#A + #B – #(AB)]
all 5-card hands from 52 cards
all 5-card hands from the 48 non-king cards
all 5-card hands from the 48 non-ace cards
all 5-card hands from the 44 non-ace and non-king cards
2/16/2019
MATH 106, Section 16

13. A = the set of arrangements where the letters E, F are together #3
Let the set of all different possible arrangements (permutations) of the letters EFNOPZ be the universal set. We define the subsets
A = the set of arrangements where the letters E, F are together
B = the set of arrangements where the letters N, O, P are together
(a)
(b)
Find the size of the universe.
#U =
6! = 720
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the letters E, F are not together
the set of arrangements where the letters N, O, P are not together
2/16/2019
MATH 106, Section 16

14. A = the set of arrangements where the letters E, F are together
Let the set of all different possible arrangements (permutations) of the letters EFNOPZ be the universal set. We define the subsets
A = the set of arrangements where the letters E, F are together
B = the set of arrangements where the letters N, O, P are together
(a)
(b)
Find the size of the universe.
#U =
6! = 720
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the letters E, F are not together
the set of arrangements where the letters N, O, P are not together
the set of arrangements where the letters E, F are together and the letters N, O, P are together
2/16/2019
MATH 106, Section 16

15. Find each of the following: #A = #B = 5!  2! = 240 4!  3! = 144
the set of arrangements where the letters E, F are not together or the letters N, O, P are not together
the set of arrangements where the letters E, F are together or the letters N, O, P are together
the set of arrangements where the letters E, F are not together and the letters N, O, P are not together
(c)
(d)
Find each of the following:
#A = #B =
5!  2! =
240
4!  3! =
144
Find the number of arrangements where the letters E, F are together and the letters N, O, P are together.
#(AB) =
3!  2!  3! = 72
2/16/2019
MATH 106, Section 16

16. (e)
(f)
Find the number of arrangements where the letters E, F are together or the letters N, O, P are together.
#(AB) =
#A + #B – #(AB) =
– 72 =
312
Find the number of arrangements where the letters E, F are not together and the letters N, O, P are not together.
#~(AB) =
#U – #(AB) =
720 – 312 =
408
2/16/2019
MATH 106, Section 16

17. B = the set of all arrangements where the letters O,P,Q,R are together #4
Let the set of all different possible arrangements (permutations) of the letters EFGHIOPQRUVW be the universal set. We define the subsets
A = the set of all arrangements where the letters E,F,G,H,I are together
B = the set of all arrangements where the letters O,P,Q,R are together
C = the set of all arrangements where the letters U,V,W are together
(a)
(b)
Find the size of the universe.
#U =
12! = 479,001,600
Describe in words each of the sets listed. ~A ~B ~C
set of arrangements where letters E, F, G, H, I are not together
set of arrangements where letters O, P, Q, R are not together
2/16/2019
MATH 106, Section 16

18. B = the set of all arrangements where the letters O,P,Q,R are together
Let the set of all different possible arrangements (permutations) of the letters EFGHIOPQRUVW be the universal set. We define the subsets
A = the set of all arrangements where the letters E,F,G,H,I are together
B = the set of all arrangements where the letters O,P,Q,R are together
C = the set of all arrangements where the letters U,V,W are together
(a)
(b)
Find the size of the universe.
#U =
12! = 479,001,600
Describe in words each of the sets listed. ~A ~B ~C
set of arrangements where letters E, F, G, H, I are not together
set of arrangements where letters O, P, Q, R are not together
set of arrangements where letters U, V, W are not together
2/16/2019
MATH 106, Section 16

19. AB
AC
BC
ABC
ABC
the set of arrangements where the letters E, F, G, H, I are together and the letters O, P, Q, R are together
the set of arrangements where the letters E, F, G, H, I are together and the letters U, V, W are together
the set of arrangements where the letters O, P, Q, R are together and the letters U, V, W are together
the set of arrangements where the letters E, F, G, H, I are together and the letters O, P, Q, R are together and the letters U, V, W are together
the set of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together
2/16/2019
MATH 106, Section 16

20. Find each of the following: #A = #B =
#C = #(AB) =
#(AC) = #(BC) =
#(ABC) =
8!  5! = 4,838,400
9!  4! = 8,709,120
10!  3! = 21,772,800
5!  5!  4! = 345,600
6!  5!  3! = 518,400
7!  4!  3! = 725,760
3!  5!  4!  3! = 103,680
Find the number of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together.
2/16/2019
MATH 106, Section 16

21. (d)
Find the number of arrangements where the letters E, F, G, H, I are together or the letters O, P, Q, R are together or the letters U, V, W are together.
#(ABC) =
#A + #B + #C
– #(AB)
– #(AC)
– #(BC)
+ #(ABC)
= 4,838, ,709, ,772,800
– 345,600 – 518,400 – 725, ,680 =
33,834,240
2/16/2019
MATH 106, Section 16

22. do Problem #8 in the Section 16 Homework:
For next class,
do Problem #5 and the short essay at the end on the Section 16 Handout (to be submitted in a future class), and
do Problem #8 in the Section 16 Homework:
In Problem #8, use #U – #(AB) = #U – [#A + #B – #(AB)]
How many are both a perfect square and a perfect cube?
perfect squares
perfect cubes
12 = 1
13 = 1
22 = 4
23 = 8
32 = 9
33 = 27
How many of these are there? ( )2 = 1,000,000
How many of these are there? ( )3 = 1,000,000
As time permits, let’s begin work on Problem #5 on the Section 16 Handout. 
2/16/2019
MATH 106, Section 16

23. A = the set of arrangements where the digits 0, 1 are together #5
Let the set of all different possible arrangements (permutations) of the digits be the universal set. We define the subsets
A = the set of arrangements where the digits 0, 1 are together
B = the set of arrangements where the digits 5, 6 are together
(a)
(b)
Find the size of the universe.
#U =
10! = 3,628,800
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the digits 0, 1 are not together
the set of arrangements where the digits 5, 6 are not together
2/16/2019
MATH 106, Section 16

24. A = the set of arrangements where the digits 0, 1 are together
Let the set of all different possible arrangements (permutations) of the digits be the universal set. We define the subsets
A = the set of arrangements where the digits 0, 1 are together
B = the set of arrangements where the digits 5, 6 are together
(a)
(b)
Find the size of the universe.
#U =
10! = 3,628,800
Describe in words each of the sets listed. ~A ~B
AB
the set of arrangements where the digits 0, 1 are not together
the set of arrangements where the digits 5, 6 are not together
the set of arrangements where the digits 0, 1 are together and the digits 5, 6 are together
2/16/2019
MATH 106, Section 16

25. Find each of the following: #A = #B = 9!  2! = 725,760 9!  2! =
the set of arrangements where the digits 0, 1 are not together or the digits 5, 6 are not together
the set of arrangements where the digits 0, 1 are together or the digits 5, 6 are together
the set of arrangements where the digits 0, 1 are not together and the digits 5, 6 are not together
(c)
(d)
Find each of the following:
#A = #B =
9!  2! =
725,760
9!  2! =
725,760
Find the number of arrangements where the digits 0, 1 are together and the digits 5, 6 are together.
#(AB) =
8!  2!  2! =
161,280
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MATH 106, Section 16

26. (e)
(f)
Find the number of arrangements where the digits 0, 1 are together or the digits 5, 6 are together.
#(AB) =
#A + #B – #(AB) =
725, ,760 – 161,280 =
1,290,240
Find the number of arrangements where the digits 0, 1 are not together and the digits 5, 6 are not together.
#~(AB) =
#U – #(AB) =
3,628,800 – 1,290,240 =
2,338,560
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MATH 106, Section 16

27. A = the set of integers where the digit 5 does not appear #6
Let the set of all integers from to be the universal set, where we use leading zeros as needed, i.e., we write integers such as 538 as We define the subsets
A = the set of integers where the digit 5 does not appear
B = the set of integers where the digit 6 does not appear
(a)
(b)
Find the size of the universe.
#U =
106 = 1,000,000
Describe in words each of the sets listed. ~A ~B
the set of integers where the digit 5 appears at least once
the set of integers where the digit 6 appears at least once
2/16/2019
MATH 106, Section 16

28. the set of integers where neither of the digits 5 and 6 appear
AB
~(AB)
AB
~(AB)
the set of integers where neither of the digits 5 and 6 appear
the set of integers where at least one of the digits 5 and 6 appears at least once
the set of integers where at least one of the digits 5 and 6 does not appear
the set of integers where each of the digits 5 and 6 appears at least once
(c)
Find each of the following:
#A = #B =
96 = 531,441
96 = 531,441
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MATH 106, Section 16

29. #U – #(A  B) = #U – [#A + #B – #(A  B)] =
(d)
(e)
Find the number of integers where neither of the digits 5 and 6 appear.
#(AB) =
86 = 262,144
Find the number of integers where the digits 5 and 6 each appears at least once.
#~(A  B) =
#U – #(A  B) = #U – [#A + #B – #(A  B)] =
106 – [ – 86] = 199,262
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MATH 106, Section 16

30. A = the set of integers divisible by 10 #7
Let the set of all integers from 1 to 1000 be the universal set. We define the subsets
A = the set of integers divisible by 10
B = the set of integers divisible by 15
C = the set of integers divisible by 25
(a)
(b)
Find the size of the universe.
#U =
1000
Describe in words each of the sets listed. ~A ~B ~C
set of integers not divisible by 10
set of integers not divisible by 15
set of integers not divisible by 25
2/16/2019
MATH 106, Section 16

31. the set of integers divisible by both 10 and 15
AB
AC
BC
ABC
ABC
the set of integers divisible by both 10 and 15
the set of integers divisible by both 10 and 25
the set of integers divisible by both 15 and 25
the set of integers divisible by each of 10, 15, and 25
the set of integers divisible by at least one of of 10, 15, and 25
2/16/2019
MATH 106, Section 16

32. Find each of the following: #A =
#B =
#C =
#(AB) =
100
110 = 10, 210 = 20, 310 = 30, …,
10010 = 1000 66
115 = 15, 215 = 30, 315 = 45, …,
6615 = 990, 6715 = 1005 40
125 = 25, 225 = 50, 325 = 75, …,
4025 = 1000 33
10 = 25
15 = 35
Integers divisible by each of 10 and 15 must be divisible by
235 = 30
130 = 30, 230 = 60, 330 = 90, …,
3330 = 990, 3430 = 1020
2/16/2019
MATH 106, Section 16

33. Integers divisible by each of 10 and 25 must be divisible by 255 =
#(BC) =
#(ABC) = 20
10 = 25
25 = 55
Integers divisible by each of 10 and 25 must be divisible by
255 = 50
150 = 50, 250 = 100, 350 = 150, …,
2050 = 1000 13
15 = 35
25 = 55
Integers divisible by each of 15 and 25 must be divisible by
355 = 75
175 = 75, 275 = 150, 375 = 225, …,
1375 = 975, 1475 = 1050 6
10 = 25
15 = 35
25 = 55
Integers divisible by all of 10, 15, and 25 must be divisible by
2355 =
150
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MATH 106, Section 16

34. (d)
(e)
Find the number of integers which are divisible by all of 10, 15, or 25.
#(ABC) = 6
Find the number of integers which are divisible by at least one of 10, 15, or 25.
#(ABC) =
#A + #B + #C
– #(AB)
– #(AC)
– #(BC)
+ #(ABC) =
– 33 – 20 – 13 +
6 =
146
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MATH 106, Section 16

35. (f)
Find the number of integers which are divisible by none of 10, 15, or 25.
#~(ABC) =
#U – #(ABC) =
1000 – 146 = 854
2/16/2019
MATH 106, Section 16

36. For next class, do the following problems in the Section 16 Homework:
integers divisible by 15
integers divisible by 12
integers divisible by 18
Problem #6: Use #U – #(ABC) =
#U – [#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)]
You will need to know that 12 = 223,
15 = 35,
18 = 233.
integers with no digit 4
integers with no digit 3
integers with no digit 5
Problem #7: Use #U – #(ABC) =
#U – [#A + #B + #C – #(AB) – #(AC) – #(BC) + #(ABC)]
2/16/2019
MATH 106, Section 16