set theory - umd department of computer science · 2016-07-13 · outline 1 branches of set theory...

Set Theory

Jason Filippou

CMSC250 @ UMCP

06-20-2016

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 1 / 56

Outline

1 Branches of Set Theory

2 Basic DefinitionsSingle setsTwo or more sets

3 Proofs with sets

4 An application: Formal languages

5 Paradoxes in Set Theory


Branches of Set Theory




Naive

Naive set theory is typically taught even at elementary schoolnowadays.

Only kind of set theory till the 1870s!Consists of applications of Venn Diagrams.

Very intuitive, suitable for graphical applications

Not an ounce of formality.

Cannot be used for...

Formal Proofs!



Naive

Naive set theory is typically taught even at elementary schoolnowadays.

Only kind of set theory till the 1870s!Consists of applications of Venn Diagrams.

Very intuitive, suitable for graphical applications

Not an ounce of formality.

Cannot be used for... Formal Proofs!



Naive

Based entirely on Venn Diagrams.

Ω

Α ΒC

Figure 1: An example Venn Diagram.



Axiomatic (Cantor & Dedekind)

First axiomatization of Set Theory.

Understanding of infinite sets and their cardinality.

Figure 2: Georg Cantor, 1870s Figure 3: Richard Dedekind, 1900s



Famous Result

1874 Cantor paper: “On a Property of the Collection of All RealAlgebraic Numbers”

The set of real numbers is uncountable.

Also:

Power set operation.Cantor’s paradise.



Russel’s Paradox

Consider the following set:

S = x|x /∈ x

Then, does S ∈ S ?



So why do we care?

If Axiomatic (Cantorian) Set Theory is “broken”, why do westudy it?

Rough answer: Just because a theory is “broken” (i.e leads tocontradictions) doesn’t mean we shouldn’t study it.

Theories are specialized (more stuff is added to them) in order toavoid contradictions all the time.

Non-Euclidean geometries.Zermello - Fraenkel Set Theory.

Qualitative answer: it gives us background necessary to discuss:1 Limitations of computers as a whole!2 Some fundamental results on countability and uncountability of

infinite sets.



Branches of Axiomatic Set Theory

Zermelo-Fraenkel (answers Russel’s paradox through the axiomof choice).

Kripke-Platek

Von Neumann - Bernays - Godel

Morse-Kelley

Tarski-Grothendieck

. . .


Basic Definitions

Basic Definitions


Basic Definitions Single sets

Single sets



Definitions

Definition (Set)

A set is a collection of objects without repetitions.

Definition (Ordered Set)

An ordered set is a pair (S,≤), where S is a set and ≤ is a totalorder.

Total orders: binary relations that are antisymmetric,transitive and total.

Examples: ≤, ⊆, lexicographic ordering.

Definition (Multiset)

A multiset is a collection of objects with repetitions.

We won’t really care about those, or about ordered multisets.



Membership

Chief operation on sets: membership (∈).

If Ω is a domain of choice and S is a set, any element e of Ω caneither belong to A (e ∈ A) or not (e /∈ A).

Since the chief operation is membership, and sets have uniqueelements, how would you implement them in computer memory?



Membership

Chief operation on sets: membership (∈).

If Ω is a domain of choice and S is a set, any element e of Ω caneither belong to A (e ∈ A) or not (e /∈ A).Since the chief operation is membership, and sets have uniqueelements, how would you implement them in computer memory?



Defining a set

Three ways.

1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .

2 Definition: A = z ∈ Z | z ≥ −23 Agreed upon symbol: N, P, etc4 An operation (union, superset, etc): C = A ∪B, P(0, 1)



Defining a set

Three ways.1 Curly braces: S = 0, 2, 4, 6, Z = Ashley, John,Mark,F = 1, 2, 3, 5, 8, 13, 21, . . .




Defining a set


2 Definition: A = z ∈ Z | z ≥ −2

3 Agreed upon symbol: N, P, etc4 An operation (union, superset, etc): C = A ∪B, P(0, 1)



Defining a set


2 Definition: A = z ∈ Z | z ≥ −23 Agreed upon symbol: N, P, etc

4 An operation (union, superset, etc): C = A ∪B, P(0, 1)



Defining a set





Cardinality

Definition (Cardinality of a set)

Let A be a finite set.a Then, the number of elements of the A, denoted|A| ∈ N is called the cardinality of A.

aHold your horses, please.

Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ?

3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ?

100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ?

99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ?

ℵ0 (???)



Cardinality




Corollary

∀A ⊆ Ω, |A| ≥ 0

|−10, 0, 10| = ? 3

|n ∈ N|n < 100| = ? 100

|n ∈ N∗|n < 100| = ? 99

|N| = ? ℵ0 (???)



Infinite sets

Certain sets are infinite!

Examples: N, P, (a, b, c) ∈ N+|a2 + b2 = c2Their cardinality cannot be expressed in the same way as that ofa finite set.

To talk about cardinality of infinite sets, we need some functionbackground...Namely, bijections, injections, surjections....



The empty set

Definition (Empty set)

There exists a unique set with no elements, denoted ∅ or and calledthe empty set.

Corollary

|∅| = 0

Corollary

∀A ⊆ Ω, ∅ ⊆ A



Some practice

How many elements do the following sets contain?

1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅

2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅2 ∅

3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅2 ∅3

4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . .

5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . .

6 . . . 250 . . . 7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . .

7 ∅, ∅



Some practice

How many elements do the following sets contain?1 ∅2 ∅3 4 . . . ∅ . . . 5 . . . . . . 6 . . . 250 . . . 7 ∅, ∅



The powerset

Definition (Powerset)

Let A be a set. The powerset of A, denoted P(A), is the set of allsubsets of A.

Corollary

∀A ⊆ Ω, ∅ ∈ P(A)

Corollary

∀A ⊆ Ω, A ∈ P(A)



Powerset examples

Examples:

P(a, b, c) = ?

∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, c

P(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ?

∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1

P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ?

∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . .

P(∅) = ? ∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ?

∅P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅

P(P(a, b)) = ? Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ?

Homework!



Powerset examples

Examples:

P(a, b, c) = ? ∅, a, b, c, a, b, b, c, a, c, a, b, cP(−1) = ? ∅, −1P(N) = ? ∅, 1, 2, . . . , 1, 22, 3, . . . P(∅) = ? ∅P(P(a, b)) = ? Homework!



(Absolute) Set complement

Figure 4: Venn Diagramillustrating the absolutecomplement of a set.

Definition (Set Complement)

Let A be a set in the universaldomain Ω. The absolutecomplement of A, denoted A′, isdefined as the set: x ∈ Ω | x /∈ A.

Corollary (Complement of emptyset)

∅′ = Ω

Corollary (Complement ofuniversal domain)

Ω′ = ∅



Caution: Sets and intervals

This: 0, 1, 2, . . . , 10

Is not the same as this: [0, . . . , 10].

Intervals are sets, but sets are not intervals.



Caution: Sets and intervals

This: 0, 1, 2, . . . , 10Is not the same as this: [0, . . . , 10].

Intervals are sets, but sets are not intervals.



Exercises

Provide the following sets in interval or set notation. Assume thatΩ = R

(Z∗−)′ = ?(−1, 1] ∪ −1 = ?[0, 10]′ = ?


Basic Definitions Two or more sets

Two or more sets



Subset

Figure 5: A subset of a set

Definition (Subset)

A set B is a subset of set A,denoted A ⊆ B, if and only if∀x ∈ Ω, x ∈ B ⇒ x ∈ A

Examples:

N ⊆ ZP ⊆ Qa ∈ Z

∣∣4|a ⊆ Zeven

Definition (Superset)

A set A is a superset of set B,denoted A ⊇ B, if and only ifB ⊆ A.



Subset


Definition (Subset)


Examples:

N ⊆ Z

P ⊆ Qa ∈ Z

∣∣4|a ⊆ Zeven





Subset


Definition (Subset)


Examples:

N ⊆ ZP ⊆ Q

a ∈ Z∣∣4|a ⊆ Zeven





Subset


Definition (Subset)


Examples:

N ⊆ ZP ⊆ Qa ∈ Z

∣∣4|a ⊆ Zeven





Proper subset

Definition (Proper subset)

A set A is a proper subset of set B, denoted A ⊂ B, if it is a subsetof set B and ∃x ∈ B : x /∈ A.

Examples:

Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅

Definition (Proper superset)

A set A is a proper superset of set B, denoted A ⊃ B, if and only ifB ⊂ A.



Proper subset



Examples:

Q ⊂ R

1 ⊂ 0, 1 ⊂ ∅, ∅





Proper subset



Examples:

Q ⊂ R1 ⊂ 0, 1

⊂ ∅, ∅





Proper subset



Examples:

Q ⊂ R1 ⊂ 0, 1 ⊂ ∅, ∅





Corollaries of subset definition

Corollary (Any set is a subset of itself)

∀S ⊆ Ω, S ⊆ S

Corollary (Any set is an element of its powerset)

∀S ⊆ Ω, S ∈ P(S)

Corollary

∀A,B such that A ⊂ B, A ⊆ B

Corollary

∀S, ∅ ⊆ S

Can we prove the latter corollary?



Subset and membership

Apply care when distinguishing between membership and subsetrelationships, particularly when dealing with sets of sets.

Are the following statements true or false?

1 ⊆ 1, 2, 3 ?

T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T

1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ?

F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F

1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ?

T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T

1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ?

F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F

1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ?

T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T

∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ?

T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T

∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ?

T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T

∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ?

F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F

∅ ∈ ? T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ?

T∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T

∅ ∈ ∅ ? F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ?

F

Master those!






1 ⊆ 1, 2, 3 ? T1 ⊆ 1, 2, 3 ? F1 ⊆ 1, 2, 3 ? T1 ∈ 1, 2, 3 ? F1 ∈ 1, 2, 3 ? T∅ ⊆ Z ? T∅ ⊆ ? T∅ ∈ ? F∅ ∈ ? T∅ ∈ ∅ ? F

Master those!



Union

Figure 6: Venn Diagram illustratingthe union of two sets.

Definition (Union)

Let A and B be two sets. Then,the union between those two sets,denoted A ∪B is the setx ∈ Ω | (x ∈ A) ∨ (x ∈ B).

Examples:

−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R



Union


Definition (Union)


Examples:

−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2

−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R



Union


Definition (Union)


Examples:

−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7

a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R



Union


Definition (Union)


Examples:

−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, c

R−Q ∪Q = R



Union


Definition (Union)


Examples:

−2,−8, 1 ∪ 0, 2 =−8,−2, 1, 0, 2−4, 6,7 ∪ −10,7 =−10,−4, 6,7a,b, c ∪ b, c =a, b, b, cR−Q ∪Q = R



Corollaries of union definition

Corollary (Union is reflexive)

For every set A, A ∪A = A

Corollary (Union of a set and its absolute complement)

For every set A, A ∪A′ = Ω

Corollary (Empty set is the neutral element of union)

For every set A, A ∪ ∅ = A

Since A ∪A = A ∪ ∅ = A, why don’t we call A a neutral element ofunion as well?



Intersection

Figure 7: Venn Diagram illustratingthe intersection of two sets.

Definition (Intersection)

Let A and B be two sets. Then,the intersection between thosetwo sets, denoted A ∩B is the setx ∈ Ω | (x ∈ A) ∧ (x ∈ B).

Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅



Intersection




Examples:

12,−8, 11 ∩ 11, 2 = 11

1, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅



Intersection




Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅

∩ = ? ∅ (!)Z− ∩P = ∅



Intersection




Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)

Z− ∩P = ∅



Intersection




Examples:

12,−8, 11 ∩ 11, 2 = 111, 3, 5, 8, 13, . . . ∩−10,−20,−30 . . . = ∅ ∩ = ? ∅ (!)Z− ∩P = ∅



Corollaries of intersection definition

Corollary (Reflexivity of intersection)

For all sets A, A ∩A = A

Corollary (Intersection and subset)

For all sets A, B such that A ⊆ B, A ∩B = A.

Corollary (Identity law)

∀A ⊆ Ω, A ∩ Ω = A

Corollary (Domination law)

∀A ⊆ Ω, A ∩ ∅ = ∅



(Relative) Complement

Figure 8: Venn Diagram illustratingthe relative complement A−Bbetween two sets.

Definition (Relative complement)

Let A and B be two sets. Then,the relative complement of Awith respect to B between thosedenoted A−B or A \B is the setx ∈ Ω | (x ∈ A) ∧ (x /∈ B).

Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅







Examples:

12,−8, 11 − 11, 2 =12,−8

1, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅







Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 13

0, 1 − −1, 0, 1 = ? ∅







Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ?

∅







Examples:

12,−8, 11 − 11, 2 =12,−81, 3, 5, 8, 13, . . . −−10,−20,−30 . . . =1, 3, 5, 8, 130, 1 − −1, 0, 1 = ? ∅



Disjoint sets

Definition (Disjoint sets)

Two sets A and B are called disjoint if and only if A ∩B = ∅

Corollary

∀A ⊆ Ω, A and ∅ are disjoint.

Corollary

∀A ⊆ Ω, A and A′ are disjoint.

Corollary

For all sets A, B, A−B and B are disjoint.

Corollary

∅ is the only set disjoint from Ω.



Partition

Figure 9: Venn Diagram illustrating apartition of a set.

Definition (Partition of a set)

Let A be a set. The partition of Ais a set of sets A1, A2, . . . , An whichhave the following properties:

1 Ai and Aj are disjoint forevery i 6= j, 1 ≤ i, j ≤ n

2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .



Partition





2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+

−10,−5, 6 and−10,−5, 6 .



Partition





2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+−10,−5, 6 and−10,−5, 6

.



Partition





2⋃nk=1Ak = A.

Examples:

R and Z−, 0,N+−10,−5, 6 and−10,−5, 6 .



Ordered n-tuples

Definition (Ordered n-tuple)

Let ≤ be some ordering of elements in Ω.a For n ∈ N, (x1,x2, . . . ,xn) is an orderedn-tuple (abbrv. tuple) if x1 ≤ x3 ≤ · · · ≤ xn.

aE.g a total order.

Definition (n-tuple equality)

Let n, r ∈ N. Two tuples (x1, x2, . . . , xn), (y1, y2, . . . , yr) are equal iff:

1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES3 (−8

√2,−16,−7, 0, 1) ? NO

4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ?

YES2 (−4.5,−3.5,−2.5) ? YES3 (−8

√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES

2 (−4.5,−3.5,−2.5) ? YES3 (−8

√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ?

YES3 (−8

√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.

Are the following tuples according to the standard real number ordering ‘ ≤′?1 (2, 8, 10, 34) ? YES2 (−4.5,−3.5,−2.5) ? YES

3 (−8√

2,−16,−7, 0, 1) ? NO4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ?

NO4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO

4 (0) ?

YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO

4 (0) ? YES

5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO

4 (0) ? YES5 () ?

YES (Empty tuple)6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO

4 (0) ? YES5 () ? YES (Empty tuple)

6 (2, 3, 3, 5) ? YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO

4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ?

YES7 (0, 0.5,

√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO

4 (0) ? YES5 () ? YES (Empty tuple)6 (2, 3, 3, 5) ? YES

7 (0, 0.5,√

2, 3, 4,√

9) NO8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9)

NO8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . )

NO



Ordered n-tuples



aE.g a total order.



1 n = r.

2 ∀i = 1, 2, . . . , n, xi = yi.


√2,−16,−7, 0, 1) ? NO


√2, 3, 4,

√9) NO

8 (0, 1, 2, . . . ) NOJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 37 / 56


Cartesian Product

Definition (Cartesian Product)

Let A and B be sets. Then, the Cartesian Product of A and B,denoted A×B, is the set: (a, b)|a ∈ A, b ∈ B

Examples:

0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .

Order matters! (A×B) 6= (B ×A)

The elements of the Cartesian Product are ordered n-tuples!



Cartesian Product



Examples:

0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)

Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .





Cartesian Product



Examples:

0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)

N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .





Cartesian Product



Examples:

0, 1 × 3, 4 = (0, 3), (0, 4), (1, 3), (1, 4)Suppose A = Rachel,Mary,Katherine and B = Rick, Chris,then A×B =(Rachel, Rick), (Rachel, Chris), (Mary,Rick), (Mary,Chris),(Katherine,Rick), (Katherine, Chris)N× N =(0, 0), (0, 1), (0, 2), . . . , (1, 0), (1, 1), (1, 2), . . . , (2, 0), (2, 1), . . .




Proofs with sets

Proofs with sets


Proofs with sets

Proving subset relationships

One needs to prove that whenever an element belongs to a set x, itmust belong to the other.

Examples:

N ⊆ Qs | s is a student registered in 250 ⊆ UMD StudentsKansas Counties ⊆ USA Counties

What kinds of proofs are required here?


Proofs with sets

An example proof

Theorem

For any sets A and B, A ∩B ⊆ A

Proof.

Let M , N be generic particular sets. Then, by the definition ofintersection, M ∩N = x|x ∈M ∧ x ∈ N . So all those elements xbelong in M as well, which means that M ∩N ⊆M by the definitionof subset. Since M and N were chosen arbitrarily, the result holds forevery pair of sets A,B.


Proofs with sets

Proving equality relationships

Definition (Set equality)

Let A and B be sets. Then, A and B are equal, denoted A = B, if andonly if A ⊆ B and B ⊆ A.

So we need to prove the subset relationship both ways.

Examples:

Let A = Neven and B = (Z− Zodd) ∩ R+ Prove that A = B.Let A = n2 |n is odd and Zodd. Is A = B?


Proofs with sets

Take 5

Let’s split into teams and try to prove the following:1 A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)2 (A ∪B)′ = A′ ∩B′3 A ∪ (A ∩B) = A


Proofs with sets

Proving that a set is empty

Usually done via contradiction.

Assume it is non-empty, so it must contain some element x, reacha contradiction.

Prove that Zeven and Zodd are disjoint for practice.


Proofs with sets

Axioms of Classical Set Theory

For sets A, B and universal domain Ω:Commutativity A ∪B = B ∪A A ∩B = B ∩AAssociativity of union& intersection

(A ∩B) ∩ C = A ∩ (B ∩ C) (A ∪B) ∪ C = A ∪ (B ∪ C)

Distributivity ofunion & intersection

A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)

Identity laws A ∩ Ω = A A ∪ ∅ = AInverse laws A ∪A′ = Ω A ∩A′ = ∅Double complemen-tation

(A′)′ = A

Idempotence A ∩A = A A ∪A = ADe Morgan’s axioms (A ∩B)′ = A′ ∪B′ (A ∪B)′ = A′ ∩B′

Universal bound(Domination) laws

A ∪ Ω = Ω A ∩ ∅ = ∅

Absorption laws A ∪ (A ∩B) = A A ∩ (A ∪B) = AAbsolute Comple-ments of empty set /domain

∅′ = Ω Ω′ = ∅

Unnamed #1 A ⊆ B ⇒ A ∩B = A A ⊆ A ∪B = BUnnamed #2 A ⊆ B ⇒ B′

Unnamed #3 A⊕B = A ∪B −A ∩B


Proofs with sets

Applying the axioms

Now that we have the axioms of set theory set up, we can usethem to derive new relationships!

Let’s use the axioms to prove the following:

((A1 ∪A2) ∪A3) ∪A4 = A1 ∪ ((A2 ∪A3) ∪A4)(A ∪B)− C = (A− C) ∪ (B − C)

As with the propositional logic exercises, be meticulous!


An application: Formal languages




Alphabets

Any finite set can be considered an alphabet (sometimes called avocabulary).

Examples:

English alphabet: a, b, c, . . . , zGreek alphabet: α, β, . . . , ωBinary alphabet: 0, 1

Denoted Σ.



Strings

Any ordered n-tuple of symbols over an alphabet Σ.

Usually denoted σ.

Length of a string: `(σ) = the number of “characters” in σ.

Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):

acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):a

caaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):ac

aaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaa

bababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababaz

jasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjason

madagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascar

charliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharlie

thissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishletters

ε: The empty string

`(ε) = 0.



Strings


Usually denoted σ.


Examples (Σ = a, b, c, . . . , z):acaaaaccccaaaabababazjasonmadagascarcharliethissentencelooksthewayitlooksbecausemyalphabetonlycontainslowercaseenglishlettersε: The empty string

`(ε) = 0.



Σn, Σ∗ and Languages

Are we straying away from sets or something?

Nope.

Some additional definitions:

Σn, for n ∈ N: The set of strings made up of elements of Σ withlength exactly n.Σ∗: The set of all strings made up of elements of Σ with finitelength.Language L over Σ: any subset of Σ∗.

Examples (whiteboard)


Paradoxes in Set Theory




The barber paradox



The barber paradox

In rural Cleanshaveville, it is illegal for guys to have facial hair.

To accomplish this, the town employs a single barber.

The barber is tasked with shaving those, and only those menwho do not shave themselves (say, at home).

Question: Who shaves the barber?



Russel’s paradox

We already know that a set can have sets as its elements.

Let’s define a set S as the set of sets that are not membersof themselves.

Symbolically: S = x | x is a set such that x /∈ xQuestion: Is S ∈ S?



Russel’s paradox



Symbolically: S = x | x is a set such that x /∈ x

Question: Is S ∈ S?



Russel’s paradox



Symbolically: S = x | x is a set such that x /∈ xQuestion: Is S ∈ S?



The Halting Problem

Perhaps our most important result.

Demonstrated by Alan Turing.

Question: Can I write a computer program that takesanother program as input and tells me if it terminates (infinite time) or not given some input?

Symbolically, our program would look like this:

PROGRAM HALTS(P , x)INPUTS: P , a program, x, a string fed as input to P .OUTPUTS: YES, if P terminates, NO, otherwise.BEGIN

Run P on xIf P terminates, output YESelse, output NO

END



The Halting Problem

Perhaps our most important result.

Demonstrated by Alan Turing.

Question: Can I write a computer program that takesanother program as input and tells me if it terminates (infinite time) or not given some input?

Symbolically, our program would look like this:

PROGRAM HALTS(P , x)INPUTS: P , a program, x, a string fed as input to P .OUTPUTS: YES, if P terminates, NO, otherwise.BEGIN

Run P on xIf P terminates, output YESelse, output NO

ENDJason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 55 / 56


The Halting Problem

Suppose that I can write HALTS.

Then, it is also possible to write a program calledHALTS TESTER, as follows:

PROGRAM HALTS TESTER(P )INPUTS: A program POUTPUTS: YES or nothing (see below)BEGIN:

If HALTS(P, P) outputs YES, then loop foreverelse, output YES

END

Paradox.

Upshot: The program HALTS does not exist!


set theory - umd department of computer science · 2016-07-13 · outline 1 branches of set theory...

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