Set Theory - University of Arkansas at Monticello

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Set Theory
Using Mathematics to Classify
Objects
2.1 The Language of Sets
Objectives:
• Define sets
• Specify sets using both listing (roster method)
and set-builder notation
• Understand when sets are well-defined
• Use the element symbol property
• Find the cardinal number of sets
• A set is collection of objects. Each object in a
set is called an element (member)of the set.
• Elements of a set may not share the same
characteristics. In this case we represent the
set by roster method which means that we
list all the elements of the set.
• If the elements of a set share the same
characteristics, then we can represent the set
by either the roster method or represent it by
using the so called “set builder notation”
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Examples: Roster method
A={2, 5, dog, ice cream, pen, 2/3, chair}
B={-3.7, -1, 0, 1, 2.5, 4.8, 5}
C={4, 1, 80, 26, 9, 10, 14, -3}
D={4, {1,3,-5},{dog, cat, 7}, Arkansas}
Here are some well known stes:
N ={1, 2, 3, 4, 5, . . .}= set of natural numbers
W={0, 1, 2, 3, 4, 5, . . .}= set of whole numbers
I= {. . . , -2, -1, 0, 1, 2, . . . }= set of all integers
• Set-builder notation:
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Examples:
A={x: x is a student in this class}
B={x: x is a male person shorter than 5 feet}
C={x: x is an African American female in this
class}
• D={x: x is a number less than 100 and divisible
by 3}
• P={x: x is a prime number}
Well-defined Sets
• A set is well-defined if we are able to tell
whether or not any particular object is an
element of the set.
• Example: Which sets are well-defined?
(a) { x : x is an Academy Award winner }
(b) { x : x is tall }
(c) {x : x eats too much}
(d) {x : x is a even number}
Empty Sets
Examples:
A = { x : x is a negative natural number }
B = { x : x is a pink elephant living in Royer }
C={x: x is a person with three eyes}
A, B and C the same. Why?
• Do  and {} mean the same thing?
–  is the empty set – a set with no members
– {} is a set with a member object, namely, the
empty set
Universal Set
• Example:
Consider female consumers living in the U.S.
The universal set is
U   x : x is a fem ale cosum er living in the U .S . 
• Example:
Consider the set of Natural numbers.
The universal set is
U = { 1, 2, 3, 4, … }
The Element Symbol
 m e a n s "is a n e le m e n t o f"
 m e a n s "is n o t a n e le m e n t o f"
Examples:
3 
 2 , 3, 4 , 5
6 
 2 , 3, 4 , 5 
Cardinal Number
of a set
We use the notation n(A) for the cardinality of a set A.
Examples:
If A={2, 4, 5, -3, 8, -7, 2.5}, the n(A)= 7, so A is finite
X 
 1, 2, 3  , 1, 4, 5  ,  3  Then
n(x)= 3, X is finite
If N is the set of all natural numbers, then N is infinite.

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