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Survey of Mathematical IdeasMath 100Chapter 2
John Rosson
Tuesday January 30, 2007
Basic Concepts of Set Theory
1. Symbols and Terminology
2. Venn Diagrams and Subsets
3. Set Operations and Cartesian Products
4. Cardinal Numbers and Surveys
5. Infinite Sets and Their Cardinalities
Power Set
The power set of set A, denoted
is the set of all subsets of A. Thus
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P( A)
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P( A) = {x | x ⊆ A}
Power SetExample
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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
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n(P({1,2,3})) = 8
Power Set
Theorem: The number of subsets of a finite set A is given by
and the number of proper subsets is given by
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n(P( A)) = 2n( A)
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2n( A) −1
Power SetSet 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} 5 25=32 31
{1,2,3,…,100} 100 2100=12676506002282294014967032
05376
1267650600228229401496
703205375
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
is the set elements of U not in A. Thus
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′ A
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′ A ={x | x ∈U and x ∉ A}
ComplementExamples. Let U={a,b,c,d,e,f,g,h,i,j,k,l}, A={a,b,c,d}. Let C D.
CD
U
U
lkjihgfedcba
AlkjihgfeA
lkjihgfeA
′⊆′=∅′∅=′
=′=′=′′
=′
},,,,,,,,,,{}{},,,,,,,{)(},,,,,,,{
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
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M ⊂ N
Numbers as Sets
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0 =∅
1 = {0} = {∅}
2 = {0,1} = {∅ ,{∅}}
3 = {0,1,2} = {∅ ,{∅},{∅ ,{∅}}}
4 = {0,1,2,3} = {∅ ,{∅},{∅ ,{∅}},{∅ ,{∅},{∅ ,{∅}}}}
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.
IntersectionThe intersection of sets A and B, denoted
is the set of elements common to both A and B.
BA∩
} and |{ BxAxxBA ∈∈=∩
IntersectionLet A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set
}3{},,3{}3,2,1{ =∩=∩=∩∅=′∩
=∩∅=∩∅
baDCABA
AAAAU
A
Sets with empty intersection are called disjoint. Thus, every set is disjoint from its complement.
UnionThe union of sets A and B, denoted
is the set of elements belonging to either of the sets.
BA∪
}or |{ BxAxxBA ∈∈=∪
UnionLet A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set
},,3,2,1{},,3{}3,2,1{ babaDC
BBA
UAA
UAU
AA
=∪=∪=∪=′∪=∪=∪∅
DifferenceThe difference of sets A and B, denoted
is the set of elements belonging to set A but not to set B.
BA−
} and |{ BxAxxBA ∉∈=−
DifferenceLet A and B be sets with A B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set
},{}3,2,1{},,3{
}2,1{},,3{}3,2,1{
babaCD
baDC
AAA
UA
AAU
AA
AA
A
=−=−=−=−
=′−∅=−
′=−=∅−∅=−∅=−∅
Ordered PairsThe ordered pair of with first component a and second component b, denoted
is defined to be the set
Thus,Note that for ordered pairs, order is important. So
),( ba
}}.,{},{{),( baaba =
. ifonly ),(),( baabba ==
In particular, },{),( aaaa ≠
. and ifonly ),(),( dbcadcba ===
Cartesian Product
The Cartesian product of sets A and B, denoted
is the set
BA×
}. and |),{( BbAabaBA ∈∈=×
Cartesian ProductLet C = {1,2,3} and D = {3,a,b}.
)},3(),,3(),3,3(
),,2(),,2(),3,2(
),,1(),,1(),3,1{(
ba
ba
baDC =×
In general, for sets A and B:
So in the example
).()()( BnAnBAn ⋅=×
.933)()()( =⋅=⋅=× DnCnDCn
Operations on SetsOperations on sets can be combined. Let A={a,b}, B={b,c}, C={c,d}, D={b,d} and E={a,c}. Calculate in list form.Working from the inside out
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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)}
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A∩B( )∪C( )−D( ) ×E
Venn DiagramsHere is the previous set calculation as a Venn diagram. The is no adequate Venn diagram for the Cartesian product.
a
bc
BA∩b
A
B
c
DCBA −∪∩ ))((b d
D
c
b
c
dCBA ∪∩ )(
b
c
d
C
De Morgan’s Laws
For and sets A and B, the complement of their intersection is the union of their complements,
and the complement of their unions is the intersection of their complements.
BABA ′∪′=′∩ )(
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(A∪B ′ ) = ′ A ∩ ′ B
De Morgan’s Laws
The set will be all the blue not in A and not in B.
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′ A ∩ ′ 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. 881-6, 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