Chapter 15: Combinatorics

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Chapter 15 Combinatorics
L15.1 Venn Diagrams
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Warmup

• Consider the integers from 1 to 10, inclusive.
• List all the integers that are
• Prime
• Odd
• Factors of 6
• List each integer that is both prime and odd.
• List each integer that is prime, odd and a factor
  of 6.
• List each integer that is not prime, not odd, and
  not a factor of 6.
• List each integer that is both odd and a factor
  of 6, but not prime.

2, 3, 5, 7
1, 3, 5, 7, 9
1, 2, 3, 6
3, 5, 7
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4, 8, 10
1
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Combinatorics Venn Diagrams

• Combinatorics the Theory of Counting
• Sets can be used to separate objects into
  containers to be counted.
• Venn Diagrams can be used to illustrate those set
  containers.
• A rectangle is used to represent a universal set,
  U
• Inside, circles represent subsets of the
  universal set.
• Overlaps of circles generate additional
  containers.

B
A
U
Complement Set of all elements not in A U B
Union Set of all ele- ments in A, B, or both
Intersection Set of all elements in both A and B
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Venn Diagrams Numbering
Outside numbers are not commonly used
In the example to the right, - the universal
set has 1000 elements. - set A has 350
elements. - set B has 140 elements.
1000
310
40
100
This is written n(U) 1000, n(A) 350,
n(B) 140
550
A number outside of a figure refers to the number
of elements in that set.
This is called the Inclusion Exclusion
Principle
There are 40 elements in AnB, n(AnB) 40.
A number inside a figure refers to the elements
only in that part of the figure.
How many elements are in A or B?
n(AUB) n(A) n(B) n(AnB)
450
350 40 310 140 40 100
n(A) n(AnB) n(B) n(AnB)
How many elements are in A alone?
in B alone?
How many elements are not in A or B?
1000 450
550
The sum of all the inside numbers n(U).
310 40 100 550 1000
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Venn Diagrams Examples
Example 1
Find n(AUB) n(A) n(B)
27 27 35 89 54 62 116
27
35
27
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Example 2 Consider the first 100 positive
integers. 25 are prime, 9 are factors of
100, and 68 are neither prime nor factors of
100. How many are a) both prime and
factors of 100? b) a factor of 100 but
not prime?
(25 x) x (9 x) 100 68
x 2
(9 x) 9 2 7
Let P be set of primes F be factors of 100.
x
9 x
25 x
Let x n(PnF).
68
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Venn Diagrams Examples
Example 3 Draw a Venn Diagram to illustrate each
of the following a) All As are Bs b) No
As are Bs c) Some As are Bs
a)
or
Note AnB ?
b)
empty set
c)
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Venn Diagrams Unions Intersections
A
B
AnB
AnC
BnC
C
AnBnC
U
n(AUBUC) n(A) n(B) n(C) n(AnB) n(AnC)
n(BnC) n(AnBnC)
Again, this is called the Inclusion Exclusion
Principle
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Venn Diagrams
Example 4 Shade the specified set a) AnB
b) An(BUC) c) BU(AnC) d) BU(AnC)
a)
b)
c)
d)
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Class Exercises
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