Download Survey of Mathematical Ideas Math 100 Chapter 2

Survey of Mathematical Ideas
Math 100
Chapter 2
John Rosson
Tuesday January 30, 2007
Basic Concepts of Set Theory
1.
2.
3.
4.
5.
Symbols and Terminology
Venn Diagrams and Subsets
Set Operations and Cartesian Products
Cardinal Numbers and Surveys
Infinite Sets and Their Cardinalities
Power Set
The power set of set A, denoted
P( A)
is the set of all subsets of A. Thus


P(A)  {x | x  A}
Power Set
Example
P({1,2,3})  { ,
{1},{2},{3},
{1,2},{1,3},{2,3},
{1,2,3} }
In particular, the number of subsets of {1,2,3} is

n(P({1,2,3}))  8
Power Set
Theorem: The number of subsets of a finite set A
is given by
n(P( A))  2n( A)
and the number of proper subsets is given by


2 n( A) 1
Power Set
Set
Cardinality
# Subsets
# Proper Subsets
Ø
0
20=1
1-1=0
{a}
1
21=2
2-1=1
{a,b}
2
22=4
4-1=3
{1,2,3}
3
23=8
7
{1,2,c,4,5}
{1,2,3,…,100}
5
100
25=32
31
2100=12676 12676506002
5060022822 28229401496
9401496703 703205375
205376
Complement
The collection of all possible element of sets, either
stated or implied, is called the universal set, often
denoted U.
For any subset A of the universal set U, the
complement of A, denoted
A
is the set elements of U not in A. Thus
A {x | x U and x  A}

Complement
Examples. Let U={a,b,c,d,e,f,g,h,i,j,k,l}, A={a,b,c,d}.
Let C  D.
A  {e, f , g , h, i, j , k , l}
( A)  {e, f , g , h, i, j , k , l}  A
{a}  {b, c, d , e, f , g , h, i, j , k , l}
U  
  U
D  C 
Venn Diagrams
A Venn diagram is a pictorial representation of
sets and their various relations and operation.
The first picture below represents the universal
set U, a set A, and the complement of A. The
second represents the relation M  N

Numbers as Sets
All mathematical objects can be defined in terms
of sets. The example below indicates how one
might define the first five whole numbers as sets.
0
1  {0}  {}
2  {0,1}  {,{}}
3  {0,1,2}  {,{},{,{}}}
4  {0,1,2,3}  {,{},{,{}},{,{},{,{}}}}
Intersection
The intersection of sets A and B, denoted
A B
is the set of elements common to both A and B.
A  B  {x | x  A and x  B}
Intersection
Let A and B be sets with A  B. Let C = {1,2,3} and D = {3,a,b}.
Let U be the universal set
 A 
U A A
A  A  
A B  A
C  D  {1,2,3}  {3, a, b}  {3}
Sets with empty intersection are called disjoint. Thus, every set is
disjoint from its complement.
Union
The union of sets A and B, denoted
A B
is the set of elements belonging to either of the sets.
A  B  {x | x  A or x  B}
Union
Let A and B be sets with A  B. Let C = {1,2,3} and D = {3,a,b}.
Let U be the universal set
 A A
U AU
A  A  U
A B  B
C  D  {1,2,3}  {3, a, b}  {1,2,3, a, b}
Difference
The difference of sets A and B, denoted
A B
is the set of elements belonging to set A but not to
set B.
A  B  {x | x  A and x  B}
Difference
Let A and B be sets with A  B. Let C = {1,2,3} and D = {3,a,b}.
Let U be the universal set
 A 
A A  
A  A
U  A  A
A U  
A  A  A
C  D  {1,2,3}  {3, a, b}  {1,2}
D  C  {3, a, b}  {1,2,3}  {a, b}
Ordered Pairs
The ordered pair of with first component a and
second component b, denoted
( a, b)
is defined to be the set
(a, b)  {{a},{a, b}}.
Thus, (a, b)  (c, d ) only if a  c and b  d .
Note that for ordered pairs, order is important. So
(a, b)  (b, a) only if a  b.
In particular,
(a, a)  {a, a}
Cartesian Product
The Cartesian product of sets A and B, denoted
A B
is the set
A  B  {( a, b) | a  A and b  B}.
Cartesian Product
Let C = {1,2,3} and D = {3,a,b}.
C  D  {(1,3), (1, a), (1, b),
(2,3), (2, a), (2, b),
(3,3), (3, a ), (3, b)}
In general, for sets A and B: n( A  B)  n( A)  n( B).
So in the example n(C  D)  n(C )  n( D)  3  3  9.
Operations on Sets
Operations on sets can be combined. Let A={a,b}, B={b,c}, C={c,d},
D={b,d} and E={a,c}. Calculate A B C D E in list form.
Working from the inside out

A B  C D E  {a,b}  {b,c}  C  D E
 {b}  {c,d}  D  E
 {b,c,d}  {b,d}  E
 {c}  {a,c}
 {(c,a),(c,c)}
Venn Diagrams
Here is the previous set calculation as a Venn diagram. The is no adequate
Venn diagram for the Cartesian product.
A
A B
b
b
c
a
B
( A  B)  C
b
b
d
c
d
c
C
D
(( A  B)  C )  D
b
d
c
c
De Morgan’s Laws
For and sets A and B, the complement of their
intersection is the union of their complements,
( A  B)  A  B
and the complement of their unions is the
intersection of their complements.
(A B) A B

De Morgan’s Laws
The set A B will be all the blue not in A and not in B.

Assignments 2.4, 2.5, 3.1
Read Section 2.4
Due February 1
Exercises p. 79
1, 3, 5, 7, 9, 17, 19, 25, and 27.
Read Section 2.5
Due February 6
Exercises p. 88
1-6, 7, 9, 11, 13, 14, 15, 24, 29,
32, 37, 38, 39, 40, 43.
Read Section 3.1
Due February 8
Exercises p. 99
1-9, 39-47, 49-53, 57-74