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Chapter 2 Basic Concepts of Set

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					Chapter 2 Basic Concepts of Set Theory
2.1 Symbols and Terminology
Set: a collection of objects
Elements (members): objects belonging to a set
There are three ways to designate a set:
1. Word description
2. Listing method
3. Set-builder notation
Ex: 1. The set of even counting numbers less than ten
    2. { 2, 4, 6, 8 }
    3. { x | x is an even counting number }
We use CAPITAL letters to denotes names of sets
usually A, B, and C.
We use lower case letters for elements of sets.
Ex: A = { a, b, c}
Empty Set (null set): is the set containing no elements.
Notation: { } or  are ways of denoting the empty set
{  } is not the empty set {0} also is not the empty set
Sets of Real Numbers
Natural (counting) numbers: { 1, 2, 3, 4, 5, …}
Whole numbers: { 0, 1, 2, 3, 4, 5, …}
Integers: { …-3, -2, -1, 0, 1, 2, 3, …}
Rational numbers: any number that can be written in the
form a b if a and b are intergers and b  0.
Ex: { 5, -4, ¾, .25, .3333…. }
Irrational numbers: number that aren’t rational.
IE: numbers that can not be expressed as fractions.
Ex: {  , e, 2 , 4 8 }
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Cardinal Number (cardinality): the number of elements
in a set; repeated elements should not be counted more
than once.
Notation: n(A) = “n of A” = is the cardinal number of set A.
Finite Set: if the cardinal number of the set can be
expressed as a whole number.
Ex: {1,2,3,4,a,b,c}
Infinite Set: if the cardinal number of a set can not be
expressed as a whole number.
Ex: The set of counting numbers
Symbol Notation
:is and element of
 :is not an element of
Ex: A={1,2,3} than 2 A and 4 A
Set Equality
Set A is equal to set B provided the following two
conditions are met:
1. Every element in set A is an element is set B
2. Every element in set B is an element in set A
Ex: A={1,2,3} B={1,2,1,2,1,2,1,3,3,3,3,3} then A = B

2.2 Venn Diagrams and Subsets
Universal set: the set of all elements to be discussed:
this is sometimes implied or directly given.
Notation: U = universal set
Venn diagram: a visual way of depicting the relationship
between sets using a rectangle to denote the universal
set and circles to denote sets with in the universal set.
The Compliment of a Set
For any set A within the universal set U, the compliment
of A is the set of elements of U that are not elements of A
Notation: A’ = compliment of set A = {x | x  U and x A}
Ex: U = { 1,2,3,4}, A={1,2}, A’ = {3,4}
Subset of a Set: Set A is a subset of set B if every
element in set a is an element of set B
Notation: A  B = “Set A is a subset of set B”
Ex: A={1,2,3} B={1,2,3,4,5,6} then A  B
Alternative Def’n for Equality of sets
Set A = Set B if A  B and B  A
Proper Subset of a Set: Set A is a proper subset of
set B if A  B and set A is NOT equal to set B.
Notation: A  B = “set A is a proper subset of set B”
Ex: A={1,2,3} B={1,2,3,4} then A  B
Finding the number of Subsets and Proper Subsets:
The number of subsets of a given set = 2 n
if n = # of elements in a set.
The number of proper subsets of a given set = 2 n - 1
if n = # of elements in a set.

2.3 Set Operations and Cartesian Products
Intersection of Sets: the intersection of sets A and B,
written “ A  B ”, is the set of all elements common to both
set A and set B, or A  B = { x | x A and x B } .
Ex: If A = {1,2,3} B = {3,4,5} then   A B = { 3 }
Union of Sets: the union of set A and set B, written
“ A  B ”, is the set of all elements belonging to either set
A or set B or both, or A  B = { x | x A or x B }.
Ex: If A = {1,2,3} and B = {2,3,4} then   A  B = {1,2,3,4}
Difference of Sets: the difference of set A and set B,
written “A – B”, is the set of all elements belonging to set
A and not to set B, or A – B = { x | x A and x B }
Ex: If A = {1,2,3,} and B = {3,4,5} then A – B = {1,2}
 Ordered Pairs: in the ordered pair (a,b), a is called the
 first component and b is called the second component. In
 general (a,b)  (b,a). Two ordered pairs are equal only if
 there first components are equal and their second
 components are equal, (a,b) = (c,d) if a = c and b = d.
                Ordered pairs are NOT sets
 Cartesian Product of Sets: the Cartesian product of
 sets A and B, written A x B, is
              A x B = {(a,b)| a A and b B}.
 Ex: Let A = {1,2,3} and B = {a,b} then
     A x B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}
 Cardinal Number of a Cartesian Product: If n(A) = a
 and n(B) = b, then n(AxB) = n(A) x n(B) = n(B) x n(A) = ab
 De Morgan’s Laws: for any sets A and B
        b g  
    1. A  B  A  B
        b g  
    2. A  B  A  B
2.4 Cardinal Numbers and Surveys
Cardinal Number Formula: for any sets A and B,
n( A  B)  n( A)  n( B)  n( A  B)
n( A  B)  n( A)  n( B)  n( A  B)
Ex: Find n(A  B) if n(A) = 3, n(B) = 4, and n(A  B) = 2
2.5 Infinite Sets and Their Carndinalities
One-to-One Correspondence: there will be a one-to-
one correspondence between two sets A and B if each
element in set A is paired with an element of set B and
Each element in set B is paired with each element in set
A.
Equivalent Sets: two sets are equivalent, written A~B, if
you can put the two sets in a one-to-one correspondence.

				
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