SET THEORY - cs.ioc.eecs.ioc.ee/ITKDM/files/5_set_theory.pdfioc.pdf Next section 1 Naive set theory De nition and notation Operations on sets 2 Cardinality The Size of a Set Cardinality

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SET THEORY

Schaum's outline Chapter 1

Rosen Sec. 2.1, 2.2

October 3, 2019

[email protected] Lecture 5 October 3, 2019 1 / 41

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Contents

1 Naive set theory

De�nition and notation

Operations on sets

2 Cardinality

The Size of a Set

Cardinality of power set


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Next section

1 Naive set theory


Operations on sets

2 Cardinality

The Size of a Set



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Next subsection

1 Naive set theory


Operations on sets

2 Cardinality

The Size of a Set



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(Naive) Set Theory

De�nition

"A set is a gathering together into a whole

of de�nite, distinct objects of our

perception or of our thought � which are

called elements of the set."

Georg Cantor

(1845�1918)


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Set notation

Sets are often denoted by the �rst few capitals: A,B,C and so forth,

whereas elements od sets by the lower case letters a,b,c etc. or

numbers.

By convention, particular symbols are reserved for the most important

sets of numbers:

∅ � empty set

N � natural numbers

Z � integers

Q � rational numbers

R � real numbers

C � complex numbers

If a is a member of the set A, then we write a ∈ A.

If a is not a member of the set A, then we write a /∈ A.


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Set notation



numbers.


sets of numbers:

∅ � empty set


Z � integers


R � real numbers





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Set notation



numbers.


sets of numbers:

∅ � empty set


Z � integers


R � real numbers





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Describing a set

Roster method: list all the elements of the set between braces, dots can be used

Examples:I V = {a,e, i ,o,u,y}I L= {a,b,c, . . . ,z}I N= {0,1,2, . . .}

Characterize the elements of the set by the property (predicates) they must satisfy to bemembers

Examples;I S = {x |x is a positive integer less than 100}I S = {x |x ∈ Z+ ∧x < 100}I P = {x |P(x)} where P(x) = true i� x is a prime number

I Q+ = {q|∃n,m ∈ Z+.q = n/m}Interval notation to describe subsets of sets upon which an order is de�ned

Examples;I [a,b] = {x |a6 x 6 b} closed intervalI [a,b) = {x |a6 x < b} half-open intervalI (a,b] = {x |a< x 6 b} half-open interval

I (a,b) = {x |a< x < b} open interval


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Describing a set









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Describing a set









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Set equality

De�nitionTwo sets A and B are equal, denoted A= B, i� they have the same elements

∀x(x ∈ A↔ x ∈ B).

Examples:

The order is not important{a,e, i ,o,u,y}= {y ,u,o, i ,e,a}

Repetitions are not important

{a,e, i ,o,u,y}= {a,a,e,e, i , i ,o,o,u,u,y ,y}


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Subsets and supersets

De�nitionThe set A is a subset of B (and B is a superset of A) i� every element of A is also an element ofB. We use the notation A⊆ B to indicate that A is a subset of the set B.

Note that A⊆ B i� the quanti�cation

∀x(x ∈ A→ x ∈ B)

is true.To show that A is not a subset of B we need only �nd one element x ∈ A with x /∈ B.To show that two sets A and B are equal, show that A⊆ B and B ⊆ A.

PropositionFor every set S ,

(i) ∅⊆ S and

(ii) S ⊆ S

De�nitionThe set A is a proper subset of B, and we write A⊂ B i� A⊆ B but A 6= B. Formally,

∀x(x ∈ A→ x ∈ B)∧∃x(x ∈ B ∧x /∈ A).


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Subsets and supersets

De�nitionThe set A is a subset of B (and B is a superset of A) i� every element of A is also an element ofB. We use the notation A⊆ B to indicate that A is a subset of the set B.

Note that A⊆ B i� the quanti�cation

∀x(x ∈ A→ x ∈ B)

is true.To show that A is not a subset of B we need only �nd one element x ∈ A with x /∈ B.To show that two sets A and B are equal, show that A⊆ B and B ⊆ A.

PropositionFor every set S ,

(i) ∅⊆ S and

(ii) S ⊆ S

De�nitionThe set A is a proper subset of B, and we write A⊂ B i� A⊆ B but A 6= B. Formally,

∀x(x ∈ A→ x ∈ B)∧∃x(x ∈ B ∧x /∈ A).


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Next subsection

1 Naive set theory


Operations on sets

2 Cardinality

The Size of a Set



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Operations on sets (1)

Union of the sets A and B, denoted A∪B, is the set of all objects that are a memberof A, or B, or both. The union of {1,2,3} and {2,3,4} is the set {1,2,3,4}.

A B

Intersection of the sets A and B, denoted A∩B, is the set of all objects that are membersof both A and B. The intersection of {1,2,3} and {2,3,4} is the set {2,3}.

BA B


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Union of the sets A and B, denoted A∪B, is the set of all objects that are a memberof A, or B, or both. The union of {1,2,3} and {2,3,4} is the set {1,2,3,4}.

A B

Intersection of the sets A and B, denoted A∩B, is the set of all objects that are membersof both A and B. The intersection of {1,2,3} and {2,3,4} is the set {2,3}.

BA B


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Operations on sets (2)Set di�erence of the sets A and B, denoted A\B, is the set of all members of A that are not

members of B. The set di�erence {1,2,3}\{2,3,4} is {1} , while, conversely,the set di�erence {2,3,4}\{1,2,3} is {4}.

A B

Complement of a (sub)set A in a universal set U , denoted A′ or Ac , is the set di�erenceU \A.

A


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Operations on sets (2)Set di�erence of the sets A and B, denoted A\B, is the set of all members of A that are not

members of B. The set di�erence {1,2,3}\{2,3,4} is {1} , while, conversely,the set di�erence {2,3,4}\{1,2,3} is {4}.

A B

Complement of a (sub)set A in a universal set U , denoted A′ or Ac , is the set di�erenceU \A.

A


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Symmetric di�erence of sets A and B, denoted A4B, is the set of all objects that are amember of exactly one of A and B (elements which are in one of the sets, butnot in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetricdi�erence set is {1,4}. It is the set di�erence of the union and theintersection, (A∪B)\ (A∩B) or (A\B)∪ (B \A).

A B

Set di�erence of the sets A and B, denoted A\B, is the set of all members of A that are notmembers of B. The set di�erence {1,2,3}\{2,3,4} is {1} , while, conversely,the set di�erence {2,3,4}\{1,2,3} is {4}.


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Symmetric di�erence of sets A and B, denoted A4B, is the set of all objects that are amember of exactly one of A and B (elements which are in one of the sets, butnot in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetricdi�erence set is {1,4}. It is the set di�erence of the union and theintersection, (A∪B)\ (A∩B) or (A\B)∪ (B \A).

A B

Set di�erence of the sets A and B, denoted A\B, is the set of all members of A that are notmembers of B. The set di�erence {1,2,3}\{2,3,4} is {1} , while, conversely,the set di�erence {2,3,4}\{1,2,3} is {4}.


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Cartesian product of A and B, denoted A×B, is the set whose members are all possibleordered pairs (a,b) where a is a member of A and b is a member of B. TheCartesian product of A= {1,2} and B = {red ,white} isA×B = {(1, red),(1,white),(2, red),(2,white)}.

Power set of a set A is the set whose members are all possible subsets of A, denotedP(A). For example, the power set of A= {1,2} isP(A) = {{},{1},{2},{1,2}}.


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Cartesian product of A and B, denoted A×B, is the set whose members are all possibleordered pairs (a,b) where a is a member of A and b is a member of B. TheCartesian product of A= {1,2} and B = {red ,white} isA×B = {(1, red),(1,white),(2, red),(2,white)}.

Power set of a set A is the set whose members are all possible subsets of A, denotedP(A). For example, the power set of A= {1,2} isP(A) = {{},{1},{2},{1,2}}.


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Set Identities (1)

Identity laws: A∪∅= A, A∩U = A

Domination laws: A∪U = U , A∩∅=∅

Idempotent laws: A∪A= A, A∩A= A

Complementation law: (A′)′ = A

Complement laws: A∪A′ = U , A∩A′ =∅

Commutative laws: A∪B = B ∪A, A∩B = B ∩A

Associative laws: A∪ (B ∪C) = (A∪B)∪C , A∩ (B ∩C) = (A∩B)∩C

Distributive laws: A∩ (B ∪C) = (A∩B)∪ (A∩C), A∪ (B ∩C) = (A∪B)∩ (A∪C)

Absorption laws: A∪ (A∩B) = A, A∩ (A∪B) = A

DeMorgan's laws: (A∪B)′ = A′ ∩B ′, (A∩B)′ = A′ ∪B ′


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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (1)












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Set Identities (2)

Laws for symmetric di�erence:

Commutativity: A4B = B4A

Associativity: (A4B)4C = A4(B4C)

Distributivity: A∩ (B4C) = (A∩B)4(A∩C)

De�nition: A4B = (A∪B)\ (A∩B)

Laws for Cartesian product:

Product with empty set: A×∅=∅, ∅×A=∅

Distributivity with ∪: A× (B ∪C) = (A×B)∪ (A×C)

Distributivity with ∩: A× (B ∩C) = (A×B)∩ (A×C)

Distributivity with \: B \C) = (A×B)\ (A×C)


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Set Identities (2)

Laws for symmetric di�erence:

Commutativity: A4B = B4A

Associativity: (A4B)4C = A4(B4C)

Distributivity: A∩ (B4C) = (A∩B)4(A∩C)

De�nition: A4B = (A∪B)\ (A∩B)

Laws for Cartesian product:

Product with empty set: A×∅=∅, ∅×A=∅

Distributivity with ∪: A× (B ∪C) = (A×B)∪ (A×C)

Distributivity with ∩: A× (B ∩C) = (A×B)∩ (A×C)

Distributivity with \: B \C) = (A×B)\ (A×C)


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Next section

1 Naive set theory


Operations on sets

2 Cardinality

The Size of a Set



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Next subsection

1 Naive set theory


Operations on sets

2 Cardinality

The Size of a Set



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The size of a set

We need to compare the size of sets when we attempt

. . . to �nd the probability of an event happening;

. . . to determine the number of solutions of a problem;

. . . to estimate complexity (need for resources) of an algorithm

or computer program;

. . . decide correctness of some certain algorithms (e.g. in

cryptography, coding etc.)

. . .


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The size of a set

De�nitionTwo sets A and B are said to be equipotent , or to have the same number of elements or thesame cardinality, written |A|= |B| or A∼ B), if there exists a one-to-one correspondencebetween the elements of A and B.


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The size of a set

De�nitionTwo sets A and B are said to be equipotent , or to have the same number of elements or thesame cardinality, written |A|= |B| or A∼ B), if there exists a one-to-one correspondencebetween the elements of A and B.

De�nitionA cardinal number, or cardinal for short, is a symbol assigned to sets the same cardinality.

The cardinal number of a set A is commonly denoted by |A|, n(A), or card(A).We will use |A|.The sequence of cardinal numbers is

0,1,2,3, . . . ,n, . . . ;ℵ0,ℵ1,ℵ2, . . . ,ℵα , . . .

|A|= 0 if A=∅|A|= n if A has the same cardinality as the �nite set {1,2, . . . ,n} for some positive integern.

|A|= ℵ0 if A has the same cardinality as the in�nite set of natural numbers N. Such a setA said to be denumerable or countably in�nite.

|A|= ℵ1 if A has the same cardinality as the set of real numbers R and it is called set ofthe cardinality of the continuum.


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Finite cardinality

A

77

50

14

2

5

B

15

12

6

0

3

|A| = |B | = 5


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Finite cardinality

A

77

50

14

2

5

B

15

12

6

0

3

|A| = |B | = 5

f : A→ B

b = f (a) =√

3(a− 2)


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Finite cardinality

A

77

50

14

2

5

B

15

12

6

0

3

C

0

2

3

|A| = |B | = 5

f : A→ B

b = f (a) =√

3(a− 2)

|A| = |B |> |C | = 3

g : B → C

c = g(b) = b MOD 4


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Finite cardinality

A

77

50

14

2

5

B

15

12

6

0

3

C

0

2

3

D

sparrow

crow

crane

ostrich

|A| = |B | = 5

f : A→ B

b = f (a) =√

3(a− 2)

|A| = |B |> |C | = 3

g : B → C

c = g(b) = b MOD 4

Let D is the ordered set:

sparrow < crow < crane < ostrich|| || || ||

D = { d1 < d2 < d3 < d4 }

h : D→ C

c = h(d) = i − 1, if d = di and i 6= 2

|A| = |B |> |D| = 4> |C |


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Finite cardinality

A

77

50

14

2

5

B

15

12

6

0

3

C

0

2

3

C1

1

D

sparrow

crow

crane

ostrich

|A| = |B | = 5

f : A→ B

b = f (a) =√

3(a− 2)

|A| = |B |> |C | = 3

g : B → C

c = g(b) = b MOD 4

Let D is the ordered set:

sparrow < crow < crane < ostrich|| || || ||

D = { d1 < d2 < d3 < d4 }

h : D→ C ′ ⊃ C

c = h(d) = i − 1, if d = di

|A| = |B |> |D| = |C ′|


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Finite cardinality

Proposition

Any proper subset A of a �nite set B (i.e. A( B) is �nite and has fewer

elements than B itself, that is

|A|< |B|.

NB!

This proposition is not valid for in�nite sets!


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Finite cardinality

Proposition

Any proper subset A of a �nite set B (i.e. A( B) is �nite and has fewer

elements than B itself, that is

|A|< |B|.

NB!

This proposition is not valid for in�nite sets!


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Countable sets

De�nition

A set that is either �nite or has the same cardinality as the set of positive

integers is called countable.

An in�nite set A is countable if and only if it is possible to list the

elements of the set in a sequence (indexed by the positive integers):

A= {a0,a1,a2, . . .}.Every in�nite set has a countably in�nite subset.

An in�nite subset of a countable set is countable.


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Countable sets

De�nition








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Countable sets

De�nition








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Countable sets (2)

Some propositions about countable sets:

1 Union of a �nite set and a countably in�nite set is countably in�nite.

2 The union of two countable sets is countable.

3 Any �nite union of countable sets is countable.

4 The union of countably many �nite sets is countable.

5 The union of countably many countable sets is countable.

6 The set of positive rational numbers is countable.

7 Any �nite Cartesian product of countable sets is countable.

8 A �nite dimensional vector space over the rational numbers is a

countable set.


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Countable sets (2)










countable set.


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Countable sets (2)










countable set.


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Countable sets (2)










countable set.


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Countable sets (2)










countable set.


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Countable sets (2)










countable set.


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Countable sets (2)










countable set.


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Countable sets (2)










countable set.


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Rational numbers

TheoremThe set of positive rational numbers is countable.

Proof.


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An uncountable set

Theorem

An interval (0,1) of real numbers is uncountable.

Proof. (Cantor diagonalization argument, � 1891). Let's suppose that the set of real numbersbetween 0 and 1 is countable. Under this assumption, all elements of the set can be listed insome order, say, r0, r1, . . .. Let the decimal representation of these real numbers be

r0 = 0,α00α01 . . .α0j . . .

r1 = 0,α10α11 . . .α1j . . .

. . . . . . . . . . . . .

ri = 0,αi0αi1 . . .αij . . .

. . . . . . . . . . . . .

Then, form a new real number with decimal expansion r = 0,β0β1 . . ., where the decimal digits

are determined so that β0 6= α00,β1 6= α11, . . . ,βj 6= αjj . . .. The number r ∈ (0,1), but it di�ers

from the decimal expansion of ri in the ith place to the right of the decimal point, for each i .

This leads to contradiction with the assumption made.


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An uncountable set (2)

Conjecture (1)

Two intervals of real numbers (0,1) and (a,b) are equipotent for any a< b.

There is one-to-one correspondence between the intervals as the linear function

y = a+(b−a)x

where x ∈ (0,1) and y ∈ (a,b).


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Conjecture (2)

the set of real numbers is uncountable.

As the intervals (0,1) and (−1,1) are equipotent, it is enough to constructone-to-one correspondence between x ∈ (−1,1) and r ∈ R. This can be doneusing equality

r =

{x/(1−x), if x ∈ [0,1);x/(1+ x), if x ∈ (−1,0).

Geometrical interpretation:

(1,1)(−1,1)

r

x

x r


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Conjecture (2)

the set of real numbers is uncountable.

As the intervals (0,1) and (−1,1) are equipotent, it is enough to constructone-to-one correspondence between x ∈ (−1,1) and r ∈ R. This can be doneusing equality

r =

{x/(1−x), if x ∈ [0,1);x/(1+ x), if x ∈ (−1,0).

Geometrical interpretation:

r

x


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Cardinality of the continuum

De�nition

The cardinality of the continuum is the cardinality or �size� of the set of

real numbers R.|R|= ℵ1.

Examples of sets with the cardinality of the continuum:

Open and closed intervals (a,b) and [a,b] of real numbers;

As well as half-open intervals (a,b] and [a,b);

The sets (Euclidean spaces) Rn, where n = 1,2,3, . . .;

Complex numbers C.


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Cardinality of the continuum

De�nition

The cardinality of the continuum is the cardinality or �size� of the set of

real numbers R.|R|= ℵ1.

Examples of sets with the cardinality of the continuum:

Open and closed intervals (a,b) and [a,b] of real numbers;

As well as half-open intervals (a,b] and [a,b);

The sets (Euclidean spaces) Rn, where n = 1,2,3, . . .;

Complex numbers C.


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Conjecture

If A is a proper subset of (in�nite) set B (A( B), then

|A|6 |B|

Conjecture

ℵ0 < ℵ1 or |N|< |R|


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Next subsection

1 Naive set theory


Operations on sets

2 Cardinality

The Size of a Set



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Theorem

For a set with n elements there are 2n subsets.

The idea of proof 1: To choose a subset one has two choices to for each element: either youput it in your subset, or you don't; and these choices are all independent. Seean example of the corresponding decision tree on next slide.

Proof 2: Induction. Base case for the empty set is true, because ∅ has exactly onesubset, namely ∅ itself. On another hand, for ∅, n= 0 and 20 = 1.

Inductive step. Now assume that the claim is true for sets with n elements.Given a set A with n+1 elements, we can write A= B ∪{p} where B is a setwith n elements and p /∈ B. There are 2n subsets X ⊆ B, and each subset givesrise to two subsets of A, namely X ∪{p} and X itself. Moreover, every subsetof A arises in this manner. Therefore the number of subsets of A is equal to2n ·2= 2n+1.


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Decision tree for setting up the power set of A= {a,b,c}

a ∈ S

b ∈ S

c ∈ S

{a,b,c}+

{a,b}−

+

c ∈ S

{a,c}+

{a}−

−

+

b ∈ S

c ∈ S

{b,c}+

{b}−

+

c ∈ S

{c}+

∅−

−

−


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Cantor's theorem

Theorem

The power set P(A) has a strictly greater cardinality than A itself.

Proof. Two sets have the same cardinality i� there exists a one-to-one correspondencebetween them. It is enough to show that, for any given set A, no functionf : A→P(A), can be surjective, i.e. to show the existence of at least onesubset of A that is not an element of the image of A under f .Let's construct the set

B = {x |x /∈ f (x)}

This means, by de�nition, that for all x in A, x ∈ B if and only if x /∈ f (x). Forall x the sets B and f (x) cannot be the same because B was constructed fromelements of A whose images (under f ) did not include themselves. Morespeci�cally, consider any x ∈ A, then either x ∈ f (x) or x /∈ f (x). In the formercase, f (x) cannot equal B because x ∈ f (x) by assumption and x /∈ B by theconstruction of B. In the latter case, f (x) cannot equal B because x /∈ f (x) byassumption and x ∈ B by the construction of B.So, we have contradiction in both cases.


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Cardinality of the power set of N

TheoremR has the same cardinality as the power set of N, i.e. P(N)∼ R.

Proof. It is enough to show that P(N)∼ [0,1).

|P(N)|6 |[0,1)| because of the mapping that assigns to every set ofnatural numbers A the real number 0. i0i1i2 . . ., where ik = 1 or ik = 0 ifk ∈ A or k /∈ A, respectively.|P(N)|6 |[0,1)| because we can assign to a number x ∈ [0,1) the set ofnatural numbers that contain k depending on whether x belongs to theleft or right half of the kth bisection of the interval [0,1).Theorem follows then due to the Schröder-Bernstein theorem.

Theorem (Schröder-Bernstein)

If A and B are sets with |A|6 |B| and |B|6 |A|, then |A|= |B|.


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Cardinality of the power set of N

TheoremR has the same cardinality as the power set of N, i.e. P(N)∼ R.

Proof. It is enough to show that P(N)∼ [0,1).

|P(N)|6 |[0,1)| because of the mapping that assigns to every set ofnatural numbers A the real number 0. i0i1i2 . . ., where ik = 1 or ik = 0 ifk ∈ A or k /∈ A, respectively.|P(N)|6 |[0,1)| because we can assign to a number x ∈ [0,1) the set ofnatural numbers that contain k depending on whether x belongs to theleft or right half of the kth bisection of the interval [0,1).Theorem follows then due to the Schröder-Bernstein theorem.

Theorem (Schröder-Bernstein)

If A and B are sets with |A|6 |B| and |B|6 |A|, then |A|= |B|.


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Continuum hypothesis (G. Cantor, 1877)

There is no set whose cardinality is strictly between that of the integers

and the real numbers.

Austrian mathematician Kurt Gödel

showed in 1940 that the continuumhypothesis cannot be disproved fromthe standard Zermelo�Fraenkel settheory (ZF), even if the axiom ofchoice is adopted (ZFC). Kurt Gödel Paul Cohen

(1906�1978) (1934�2007)

American mathematician Paul Cohen showed in 1963 that the continuumhypothesis cannot be proven from those same axioms either.


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The generalized continuum hypothesis

If an in�nite set's cardinality lies between that of an in�nite set S and that

of the power set of S , then it either has the same cardinality as the set Sor the same cardinality as the power set of S .

Conjecture

If the generalized continuum hypothesis is valid, then ℵα+1 = 2ℵα .


ioc.pdf

The generalized continuum hypothesis

If an in�nite set's cardinality lies between that of an in�nite set S and that

of the power set of S , then it either has the same cardinality as the set Sor the same cardinality as the power set of S .

Conjecture

If the generalized continuum hypothesis is valid, then ℵα+1 = 2ℵα .


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SET THEORY - cs.ioc.eecs.ioc.ee/ITKDM/files/5_set_theory.pdfioc.pdf Next section 1 Naive set theory De nition and notation Operations on sets 2 Cardinality The Size of a Set Cardinality