MAT 119 FALL 2001
MAT 119
FINITE MATHEMATICS
NOTES
PART 2 – PROBABILITY
CHAPTER 6
SETS; COUNTING TECHNIQUES
6.1
Sets
Set – collection of well-defined (distinct) objects/elements
Eg.
Set of digits
Denote by roster method
D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Denote by set-builder notation
D = { x | x is a digit}
B = {x|x is a prime number less than 20} = {2, 3, 5, 7, 11, 13, 17, 19}
Empty set/Null set (Ø or { })– set with no elements
Equality of Sets
Let A and B be two sets. A and B are equal (A = B), if and only if, they have the same
elements (both sets have must the same # of elements);
otherwise they are not equal (A  B), (they may or may not have the same number of
elements.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
Subset
Let A and B be two sets. A is a subset of B or A is contained in B (A  B), if and only if,
every element in A is also an element in B; otherwise A is not a subset of A (A  B).
Proper Subset
Let A and B be two sets.
A is a proper subset of B or A is properly contained in B (A  B), if and only if, every
element in A is also in B, but there is at least one element in B that is not in A.
Otherwise A  B
ØB
Universal Set (U) – set consisting of all elements under consideration
Any set is a subset of U
Venn diagrams – picture used to present sets and their relationships with other sets in
some universal set.
Union of Two Sets
Let A and B be two sets.
The union of A with B (A  B) - (A union B) – (A or B), - set of elements either in A or
B or in both A and B.
A  B = { x | x is in A or x is in B}
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
Intersection of Two Sets
Let A and B be two sets.
The intersection of A and B (A  B), (A intersect B), (A and B) – set of elements in both
A and B
A  B = { x | x is in A and x is in B}
Disjoint Sets – have no elements in common
AB=
Complement of a set A (Ā, or A’)– set of elements in the universal set but not in the set
Ā = { x | x is not in A}
AA  U
AA  
AA
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
De Morgan’s Properties
Let A and B be two sets. Then
a.
AB  AB
b.
AB  AB
a.
b.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
6.2
The Number of Elements in a Set
Finite set – number of elements in a set is 0 or a positive integer
A = {1, 2, 3, 4, 5}
c(A) = 5
Infinite sets – has infinitely many elements
Finite Mathematics – studies finite sets
If A and B are two finite sets, then
c(A  B) = c(A) + c(B) – c(A  B)
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
6.3
The Multiplication Principle
If we can perform a first task in p different ways, a second task in q different ways, a
third task in r different ways, …, then the total act of performing the first task, followed
by performing the second task, and so on, can be done in p.q.r. … different ways.
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
6.4
Permutations
Factorial Notation
0! = 1
1! = 1
2! = 2  1
3! = 3  2 . 1
4! = 4  3  2  1
=2
=6
= 24
In general,
n! = n  (n – 1)  (n – 2)  …  3  2  1
(n + 1)! = (n + 1)  n!
Permutations – ordered arrangements of distinct objects
r permutation of a set of n distinct objects - an ordered arrangement using r of n objects
P(n, r) – number of r permutations of a set of n distinct objects
The number of different arrangements using r objects chosen from n objects in which
1.
2.
3.
The n objects are all different.
No object is repeated in an arrangement
Order is important
is
P(n, r )  n(n  1)  ...  (n  r  1) 
n!
(n  r )!
The number of permutations (arrangements) of n different objects using
all n of them is P(n, r) = n!
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
6.5
Combinations
The number of different selections of r objects chosen from n objects in which
1.
2.
3.
The n objects are all different
No object is repeated
Order is not important
is
C (n, r ) 
n!
r!(n  r )!
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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MAT 119 FALL 2001
6.6
More Counting Problems
Permutation with repetition
The number of distinct permutations of n objects, of which n1 are of one kind, n2 of a
second kind, …, nk of a kth kind, is
n!
where n1 + n2 + … + nk = n
n1!n 2 !...  n k !
DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS
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