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