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Survey of Mathematical Ideas
Math 100
Chapter 2
John Rosson
Thursday January 25, 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
Sets
• A set is a collection of objects.
• The objects in a set are called its
elements or members.
• If A is a set and a is an element of A, we
show this in symbols as follows:
a A
a A
“a is an element of A”
“a is not an element of A”
Designating Sets
• Word description:
• Listing:
The set of positive whole numbers which are
less than 20 and evenly divisible by 7.
{7, 14}
• Set builder:
{x | x is a positive whole number and x is less than 20
and x is evenly divisible by 7}
• The empty set, designated Ø, is the set with no
elements.
Set Equality
Set A is equal to set B if
1. every element of A is an element of B and
2. Every element of B is an element of A.
This denoted, as one would expect
AB
“set A equals set B”
A B
“set A does not equal set B”
Set Equality
Examples:
{a}  {a, a}
{a, b}  {b, a}
  {}  {x | x  x}
" the set of even numbers"  " the set of numbers divisible by 2"
  {x | x is a whole number and x 2  4}
  {x | x is a whole number and x  2}
2
Sets of Numbers.
Natural or Counting numbers
Whole numbers
Integers
Rational numbers
Real numbers
Irrational numbers
{1,2,3,4,. ..}  N
{0,1,2,3,4 ,...}
{...,-3,-2,-1,0,1,2,3,...}  Z
{ p/q | p and q are integers and q  0}  Q
{x | x is a number tha t can be written as a decimal}  R
{x | x is a real number and x is not a rational number}
Cardinality
The cardinal number or cardinality of a set
is the number of element in a set.
In symbols, the cardinality of a set A is
denoted
n(A)
Equal sets have equal cardinality, but sets
with equal cardinality are not always
equal.
Cardinality
Examples
n({1,0,1})  n({1,2,3})  n({a, b, c})  3
n()  0
n({a, b, a})  n({b, a, b})  n({a, b, a, b})  n({a, b})  2
n({ x | x  N and x is odd and x  10})  5
Intuitively, the sets in the above example are
finite. On the other hand, the sets of numbers
N, Z, Q and R are all examples of infinite
sets. Later, we precise definitions of the words
“finite” and “infinite” mean.
Subsets
Set A is a subset of set B if every
element of A is also an element of B
Denoted in symbols,
A B
“set A is a subset of set B”
A / B
“set A is not a subset of set B”
If A and B are sets then A = B if A
 B and B  A.
Subsets
Set A is a proper subset of set B if A
 B and A≠B.
This denoted in symbols,
A B
“set A is a proper subset of set B”
A B
“set A is not a proper subset of set B”
Subsets
Examples. Let A be a set.
A A
 A
{1,2,3}  {1,2,3,4}
{1,2,3,4} / {1,2,3}
{1,2,3,4}  {1,2,3,...}
N Z Q R
Sets can be elements.
Any set can be an element of a set. If
A  {1,{1},2}
then

1
2
{1} 
n( A) 
A
A
A
3
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
Assignments 2.3, 2.4, 2.5
Read Section 2.3
Due January 30
Exercises p. 73
1-6, 7-27, 47, 51, 52, 71, 75, 97,
115, 127, 129, 131, 133.
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.