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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Set Theory
Jim Woodcock
January 2010
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
N i S M b hi Fi i d I fi i S E li f S E S S b P S V Di U i I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
N t ti S t M b hi Fi it d I fi it S t E lit f S t E t S t S b t P S t V Di U i I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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p q y p y g
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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p q y p y g
Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Outline
1 Notation
2 Set Membership3 Finite and Infinite Sets
4 Equality of Sets
5 Empty Set
6 Subsets7 Power Set
8 Venn Diagrams
9 Union
10 Intersection11 Difference
12 Generalised Operations
13 Russells Paradox
14 Summary
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Examples
The numbers 1, 3, 7, 4 and 10.
The solutions of the equationx2 3 x 2= 0.
The vowels of the alphabet: a, e, i, o,and u.
The people living on the earth.The studentsRobert, Catherine,andJonathan.
The students who are absent from the class.
The countriesEngland, France,andDenmark.
The capital cities of Europe.The numbers 2, 4, 6, 8, . . .
The rivers in England.
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Notation
set extension
A= {1, 3, 7, 10}
set comprehension
B={ x |x is even }
B={ x : N| x is even }
Ndenotes the set of counting numbers, 0, 1, 2, 3, . . .Convention:lower case (a,b,c) for elements, capitals (A,B,
Person,River) for set names
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Examples
A1 ={1, 3, 7, 10}
A2 ={ x : N| x2 3 x 2= 0 }
A3 ={a, e, i, o, u}
A4 ={ x :Person|x is living on the earth }A5 ={Robert, Catherine, Jonathan}
A6 ={ x :Student |x is absent from class }
A7 ={England, France, Denmark}
A8 ={ x :CapitalCity |x is in Europe }A9 ={2, 4, 6, 8, . . .}
A10={ x :River |x is in England }
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Set Membership
xA
x /A
Example
letA= {a, e, i, o, u}thena A, b /A, e A, f /A
letB={ x : N| x is even }then 3 /B, 6 B, 11 /B, 14 B
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Examples
The set of nonnegative integers less than 4.
The set of books in the Bodleian Library at the present
time.
The set consisting of people who spoke to Bertrand
Russell on the 7 June 1906.
The set of live dinosaurs in the Pitt-Rivers Museum.
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Examples
The set of integers greater than 3.
The set of all C++ programs that can be displayed on a
single screen.
The set of all C++ programs that would halt if run for a
sufficient time on a computer with unbounded storage.
The set of true propositions about the integers.
The set with two members, one of which is the set of even
integers, and other the set of odd integers.
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Finite and Infinite Sets
Examples
letMbe the set of the days of the week
thenMis finite
letN={2, 4, 6, 8, . . .}thenN is infinite
letP={ x :River |xis on Earth }although it may be difficult to count the number of rivers in
the world,Pis still a finite set
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Finite and Infinite Sets
Examples
suppose that statement labels in a (rather old-fashioned)
programming language must be either a single alphabetic
symbol or a single decimal digit
{A, B, C, . . . , Z}
{0, 1, 2, . . . , 9},
a variable name in the programming language BASICmust
be either an alphabetic symbol or an alphabetic symbol
followed by a single decimal digit
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Finite and Infinite Sets
Example
consider the four cubes puzzle
this involves four cubes each of whose faces is painted in
one of four different colours
the puzzle is to stack the cubes in such a way that each
vertical side of the stack contains squares of all four
colours
there are 3 123 =5, 184 different arrangements
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Equality of Sets
Example
{1, 2, 3, 4}={3, 1, 4, 2}{5, 6, 5, 7}={7, 5, 7, 6}
{ x : N| x2 3 x=2 }= {2, 1}={1, 2, 2, 1}
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Equality
The formal statement of when equality holds between sets is
A= B ( x :XxA xB)
ExampleSuppose that we have the following definitions:
Zeroes={ x : N| x2 3 x+ 2= 0 }SmallNums={1, 2}
Zeroes=SmallNums( x : Z
xZeroesxSmallNums)
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Empty Set
The empty set is denoted byExample
letAbe the set of people in the world who are older than
200 years
letB={ x : N| x2 =4 x is odd }
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Subsets
A B
Example
{1, 3, 5} {5, 4, 3, 2, 1}
{2, 4, 6} {6, 2, 4}
{ x : N| ( n: N x=2n) } { x : N| x is even }the set of women is a subset of the set of all humans
{1, 2, 3, 4, 5} { x : Z| 0< x
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Four Theorems
1 two setsAandBare equal if they each contain the other
A= B A BBA
2 the subset relationship isreflexive
for any setA, A A
3 since the empty set has no members, it is trivially a subset
of every other set
for any setA, A
4 the subset relation istransitive
A BBCA C
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Proper Subset
A B A BA=B
Examplethe set of natural numbers is a proper subset of the set of
integers
the set of even integers is a proper subset of the set of
integers
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Sets of Sets
it sometimes will happen that the objects of a set are sets
themselves
for example, the set of all subsets of A
the set{{2, 3}, {2}, {5, 6}}is a set of sets
its members are{2, 3}, {2}, and{5, 6}
it is not possible that a set has some members which are
sets themselves and some which arent
the type system will make sure that all the members of aset are the same kind of object
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Power Set
the family of all the subsets of any set Sis called the power
set ofS
PS
M={a, b}
PM={{a, b}, {a}, {b}, }
T ={4, 7, 8}
PT ={T, {4, 7}, {4, 8}, {7, 8}, {4}, {7}, {8}, }
A=
PA={}
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Examples
A= {1}
PA={, {1}}
A= {1, 2}
PA={, {1}, {2}, {1, 2}}
ifAis any (finite or infinite) set of natural numbers, then
APN
ifAis finite, then PAis also finite
otherwise, PAis infinite
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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Disjoint Sets
AandBaredisjointiff they have no elements in common
Examples
A= {1, 3, 7, 8}andB={2, 4, 7, 9}the set of positive numbers and the set of negative
numbers
E={x, y, z}and F ={r, s, t}
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Venn Diagrams
Example
suppose thatA B
A
B
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
E l
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Example
suppose thatAandBare not comparable
A B
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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letA={a, b, c, d}and B={c, d, e, f}
a
b
c
de
f
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
U i
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Union
we denote the union ofAwithBby
A B
Example
in the following Venn diagram, the union of the two sets A andBcontains every point marked
a
b
c
d
e
f
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Examples
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Examples
letS={a, b, c, d}andT ={f, b, d, g}, then
S T ={a, b, c, d, f, g}
letPbe the set of positive real numbers and let Qbe theset of negative real numbers
thenP Qconsists of all the real numbers except zero
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Intersection
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Intersection
the intersection of setAwith setBis the set of elements which
are common toAand toB
it is denoted by
A B
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Examples
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Examples
in the following Venn diagram, the intersection of the two
setsA andBcontains only the pointscandd:
a
b
c
d
e
f
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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letS={a, b, c, d}andT ={f, b, d, g}, then
S T ={b, d}
LetV ={2, 4, 6, . . .}, and letW ={3, 6, 9, . . .}, then
V W ={6, 12, 18, . . .}
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Difference
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Difference
the difference between two setsAandBis the set of elements
which belong toAbut not toB; we denote it by
A \ B
which is read AminusB, or AwithoutB
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Examples
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Examples
The difference between the two setsAandBcontains only
the pointsaandb:
LetS={a, b, c, d}andT ={f, b, d, g}, then
S\ T ={a, c}
a
b
c
d
e
f
letRbe the set of real numbers and let Qbe the set ofrational numbers
thenR\ Qconsists of the irrational numbers
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Formal Definitions
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Formal Definitions
A B={ x :X |xA xB}
A B={ x :X |xA xB}
A \ B={ x :X |xA x /B}
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Properties of Set Operations
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Properties of Set Operations
the operations of union and intersection are commutative and
associative
for arbitrary setsA,B, andC,
A B=B A
A B=B A
(A B) C=A (B C)
(A B) C=A (B C)
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Theorems
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Theorems
the set operations of union and intersection distribute over each
other; that is, for arbitrary sets A, B,andC,A (B C) = (A B) (A C)
A (B C) = (A B) (A C)
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
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A A= AA A= A
A = AA = A \ BAA BCDA CB DA BCDA CB D
A A BA BAA BA B=BA BA B=BA \ =AA (B\ A) =A (B\ A) =A BA \ (B C) = (A \ B) (A \ C)A \ (B C) = (A \ B) (A \ C)
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Generalised Operations
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p
A={ x :X |( S:A xS) }A={ x :X |( S:A xS) }
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Russells Paradox
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the men of a certain town can be divided into two groups:
those who shave themselvesand those who do not
the barber shaves all those who do not shave themselves
so who shaves the barber?
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Russells Paradox
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S={x :Person| (x shaves x)}= {x :Person|barber shaves x}
barberS
(barber shaves barber)barber /S
barber /S
barber shaves barber
barberS
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Russells Paradox
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S={x :Person| x shaves x}
B={x :Person|barber shaves x}Person\ S=B
barber /S
(barber shaves barber)barber /B
barberS
barberSbarber shaves barber
barberB
barber /S
Notation Set Membership Finite and Infinite Sets Equality of Sets Empty Set Subsets Power Set Venn Diagrams Union I
Summary
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sets, extension and display; finite and infinite; sets of sets
membership and non-membership /
equality: extensionality
empty set
subsets, powersets, proper subsets, disjoint sets
Venn diagrams
union, intersection, difference (minus)
generalised union and intersection
algebraic laws
axiomatic set theoryavoid the paradoxes
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