Set Theory Fundamentals

Home , Complement (set theory), Multiset, Naive set theory, Tuple, Venn diagram

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 Deﬁnitions Single sets Two or more sets

3 Proofs with sets

4 An application: Formal languages

5 Paradoxes in Set Theory

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 2 / 56 Branches of Set Theory

Branches of Set Theory

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 3 / 56 Formal Proofs!

Branches of Set Theory Naive

Naive set theory is typically taught even at elementary school nowadays. 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...

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 4 / 56 Branches of Set Theory Naive

Based entirely on Venn Diagrams.

Α Β C

Figure 1: An example Venn Diagram.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 5 / 56 Branches of Set Theory 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 6 / 56 Also: Power set operation. Cantor’s paradise.

Branches of Set Theory Famous Result

1874 Cantor paper: “On a Property of the Collection of All Real Algebraic Numbers” The set of real numbers is uncountable.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 7 / 56 Branches of Set Theory Famous Result

1874 Cantor paper: “On a Property of the Collection of All Real Algebraic Numbers” The set of real numbers is uncountable. Also: Power set operation. Cantor’s paradise.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 7 / 56 Branches of Set Theory Russel’s Paradox

Consider the following set:

S = {x|x∈ / x}

Then, does S ∈ S ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 8 / 56 Branches of Set Theory So why do we care?

If Axiomatic (Cantorian) Set Theory is “broken”, why do we study it? Rough answer: Just because a theory is “broken” (i.e leads to contradictions) doesn’t mean we shouldn’t study it. Theories are specialized (more stuﬀ is added to them) in order to avoid 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.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 9 / 56 Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck ...

Branches of Set Theory Branches of Axiomatic Set Theory

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck ...

Branches of Set Theory Branches of Axiomatic Set Theory

Zermelo-Fraenkel (answers Russel’s paradox through the axiom of choice). Kripke-Platek

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Morse-Kelley Tarski-Grothendieck ...

Branches of Set Theory Branches of Axiomatic Set Theory

Zermelo-Fraenkel (answers Russel’s paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Tarski-Grothendieck ...

Branches of Set Theory Branches of Axiomatic Set Theory

Zermelo-Fraenkel (answers Russel’s paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 ...

Branches of Set Theory Branches of Axiomatic Set Theory

Zermelo-Fraenkel (answers Russel’s paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Branches of Set Theory Branches of Axiomatic Set Theory

Zermelo-Fraenkel (answers Russel’s paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck ...

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Basic Deﬁnitions

Basic Deﬁnitions

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 11 / 56 Basic Deﬁnitions Single sets

Single sets

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 12 / 56 Deﬁnition (Ordered Set) An ordered set is a pair (S, ≤), where S is a set and ≤ is a total order.

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.

Basic Deﬁnitions Single sets Deﬁnitions

Deﬁnition (Set) A set is a collection of objects without repetitions.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56 Deﬁnition (Multiset) A multiset is a collection of objects with repetitions.

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

Basic Deﬁnitions 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 total order.

Total orders: binary relations that are antisymmetric, transitive and total. Examples: ≤, ⊆, lexicographic ordering.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56 Basic Deﬁnitions 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 total order.

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.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56 Since the chief operation is membership, and sets have unique elements, how would you implement them in computer memory?

Basic Deﬁnitions Single sets Membership

Chief operation on sets: membership (∈). If Ω is a domain of choice and S is a set, any element e of Ω can either belong to A (e ∈ A) or not (e∈ / A).

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 14 / 56 Basic Deﬁnitions Single sets Membership

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 14 / 56 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 ≥ −2} 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A ∪ B, P({0, 1})

Basic Deﬁnitions Single sets Deﬁning a set

Three ways.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 2 Deﬁnition: A = {z ∈ Z | z ≥ −2} 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A ∪ B, P({0, 1})

Basic Deﬁnitions Single sets 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,... }

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A ∪ B, P({0, 1})

Basic Deﬁnitions Single sets 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 ≥ −2}

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 4 An operation (union, superset, etc): C = A ∪ B, P({0, 1})

Basic Deﬁnitions Single sets 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 ≥ −2} 3 Agreed upon symbol: N, P, etc

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 Basic Deﬁnitions Single sets 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 ≥ −2} 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A ∪ B, P({0, 1})

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 Corollary ∀A ⊆ Ω, |A| ≥ 0

|{−10, 0, 10}| = ? 3 |{n ∈ N|n < 100}| = ? 100 ∗ |{n ∈ N |n < 100}| = ? 99 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 |{−10, 0, 10}| = ? 3 |{n ∈ N|n < 100}| = ? 100 ∗ |{n ∈ N |n < 100}| = ? 99 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 3 |{n ∈ N|n < 100}| = ? 100 ∗ |{n ∈ N |n < 100}| = ? 99 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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}| = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 |{n ∈ N|n < 100}| = ? 100 ∗ |{n ∈ N |n < 100}| = ? 99 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 100 ∗ |{n ∈ N |n < 100}| = ? 99 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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}| = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 ∗ |{n ∈ N |n < 100}| = ? 99 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 99

|N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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}| = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 |N| = ? ℵ0 (???)

Basic Deﬁnitions Single sets 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 ℵ0 (???)

Basic Deﬁnitions Single sets 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| = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 Basic Deﬁnitions Single sets 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 (???)

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 Basic Deﬁnitions Single sets Inﬁnite sets

Certain sets are infinite! + 2 2 2 Examples: N, P, {(a, b, c) ∈ N |a + b = c } Their cardinality cannot be expressed in the same way as that of a finite set. To talk about cardinality of infinite sets, we need some function background... Namely, bijections, injections, surjections....

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 17 / 56 Corollary |∅| = 0

Corollary ∀A ⊆ Ω, ∅ ⊆ A

Basic Deﬁnitions Single sets The empty set

Deﬁnition (Empty set) There exists a unique set with no elements, denoted ∅ or {} and called the empty set.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 18 / 56 Corollary ∀A ⊆ Ω, ∅ ⊆ A

Basic Deﬁnitions Single sets The empty set

Deﬁnition (Empty set) There exists a unique set with no elements, denoted ∅ or {} and called the empty set.

Corollary |∅| = 0

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 18 / 56 Basic Deﬁnitions Single sets The empty set

Deﬁnition (Empty set) There exists a unique set with no elements, denoted ∅ or {} and called the empty set.

Corollary |∅| = 0

Corollary ∀A ⊆ Ω, ∅ ⊆ A

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 18 / 56 1 ∅ 2 {∅} 3 {{}} 4 {{... {∅} ... }} 5 {{... {} ... }} 6 {{... {250} ... }} 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

How many elements do the following sets contain?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 2 {∅} 3 {{}} 4 {{... {∅} ... }} 5 {{... {} ... }} 6 {{... {250} ... }} 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

How many elements do the following sets contain? 1 ∅

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 3 {{}} 4 {{... {∅} ... }} 5 {{... {} ... }} 6 {{... {250} ... }} 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 4 {{... {∅} ... }} 5 {{... {} ... }} 6 {{... {250} ... }} 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 5 {{... {} ... }} 6 {{... {250} ... }} 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 6 {{... {250} ... }} 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 7 {∅, {∅}}

Basic Deﬁnitions Single sets Some practice

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 Basic Deﬁnitions Single sets Some practice

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 19 / 56 Corollary ∀A ⊆ Ω, ∅ ∈ P({A})

Corollary ∀A ⊆ Ω,A ∈ P({A})

Basic Deﬁnitions Single sets The powerset

Deﬁnition (Powerset) Let A be a set. The powerset of A, denoted P({A}), is the set of all subsets of A.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 20 / 56 Corollary ∀A ⊆ Ω,A ∈ P({A})

Basic Deﬁnitions Single sets The powerset

Deﬁnition (Powerset) Let A be a set. The powerset of A, denoted P({A}), is the set of all subsets of A.

Corollary ∀A ⊆ Ω, ∅ ∈ P({A})

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 20 / 56 Basic Deﬁnitions Single sets The powerset

Deﬁnition (Powerset) Let A be a set. The powerset of A, denoted P({A}), is the set of all subsets of A.

Corollary ∀A ⊆ Ω, ∅ ∈ P({A})

Corollary ∀A ⊆ Ω,A ∈ P({A})

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 20 / 56 {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}}

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}}

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... }

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 {∅} P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 P({P({a, b})}) = ? Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅}

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 Homework!

Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 Basic Deﬁnitions Single sets Powerset examples

Examples: P({a, b, c}) = ? {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}} P({−1}) = ? {∅, {−1}} P({N}) = ? {∅, {1}, {2},..., {1, 2}{2, 3},... } P({∅}) = ? {∅} P({P({a, b})}) = ? Homework!

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 21 / 56 Corollary (Complement of empty set) ∅0 = Ω

Corollary (Complement of universal domain) Ω0 = ∅

Basic Deﬁnitions Single sets (Absolute) Set complement

Deﬁnition (Set Complement) Let A be a set in the universal domain Ω. The absolute complement of A, denoted A0, is deﬁned as the set: {x ∈ Ω | x∈ / A}.

Figure 4: Venn Diagram illustrating the absolute complement of a set.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 22 / 56 Basic Deﬁnitions Single sets (Absolute) Set complement

Deﬁnition (Set Complement) Let A be a set in the universal domain Ω. The absolute complement of A, denoted A0, is deﬁned as the set: {x ∈ Ω | x∈ / A}.

Corollary (Complement of empty set) ∅0 = Ω Figure 4: Venn Diagram illustrating the absolute complement of a set. Corollary (Complement of universal domain) Ω0 = ∅

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 22 / 56 Is not the same as this: [0,..., 10]. Intervals are sets, but sets are not intervals.

Basic Deﬁnitions Single sets Caution: Sets and intervals

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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 23 / 56 Basic Deﬁnitions Single sets 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.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 23 / 56 Basic Deﬁnitions Single sets Exercises

Provide the following sets in interval or set notation. Assume that Ω = R ∗ 0 (Z−) = ? (−1, 1] ∪ {−1} = ? [0, 10]0 = ?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 24 / 56 Basic Deﬁnitions Two or more sets

Two or more sets

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 25 / 56 Examples: N ⊆ Z P ⊆ Q

{a ∈ Z 4|a} ⊆ Zeven

Deﬁnition (Superset) A set A is a superset of set B, denoted A ⊇ B, if and only if B ⊆ A.

Basic Deﬁnitions Two or more sets Subset

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

Figure 5: A subset of a set

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56 P ⊆ Q

{a ∈ Z 4|a} ⊆ Zeven

Deﬁnition (Superset) A set A is a superset of set B, denoted A ⊇ B, if and only if B ⊆ A.

Basic Deﬁnitions Two or more sets Subset

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 ⊆ Z

Figure 5: A subset of a set

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56

{a ∈ Z 4|a} ⊆ Zeven

Deﬁnition (Superset) A set A is a superset of set B, denoted A ⊇ B, if and only if B ⊆ A.

Basic Deﬁnitions Two or more sets Subset

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 ⊆ Z P ⊆ Q

Figure 5: A subset of a set

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56 Deﬁnition (Superset) A set A is a superset of set B, denoted A ⊇ B, if and only if B ⊆ A.

Basic Deﬁnitions Two or more sets Subset

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 ⊆ Z P ⊆ Q

{a ∈ Z 4|a} ⊆ Zeven Figure 5: A subset of a set

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56 Basic Deﬁnitions Two or more sets Subset

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 ⊆ Z P ⊆ Q

{a ∈ Z 4|a} ⊆ Zeven Figure 5: A subset of a set Deﬁnition (Superset) A set A is a superset of set B, denoted A ⊇ B, if and only if B ⊆ A.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 26 / 56 Q ⊂ R {1} ⊂ {0, 1} {{}} ⊂ {∅, {{∅}}}

Deﬁnition (Proper superset) A set A is a proper superset of set B, denoted A ⊃ B, if and only if B ⊂ A.

Basic Deﬁnitions Two or more sets Proper subset

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

Examples:

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56 {1} ⊂ {0, 1} {{}} ⊂ {∅, {{∅}}}

Deﬁnition (Proper superset) A set A is a proper superset of set B, denoted A ⊃ B, if and only if B ⊂ A.

Basic Deﬁnitions Two or more sets Proper subset

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

Examples: Q ⊂ R

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56 {{}} ⊂ {∅, {{∅}}}

Deﬁnition (Proper superset) A set A is a proper superset of set B, denoted A ⊃ B, if and only if B ⊂ A.

Basic Deﬁnitions Two or more sets Proper subset

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

Examples: Q ⊂ R {1} ⊂ {0, 1}

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56 Deﬁnition (Proper superset) A set A is a proper superset of set B, denoted A ⊃ B, if and only if B ⊂ A.

Basic Deﬁnitions Two or more sets Proper subset

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

Examples: Q ⊂ R {1} ⊂ {0, 1} {{}} ⊂ {∅, {{∅}}}

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56 Basic Deﬁnitions Two or more sets Proper subset

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

Examples: Q ⊂ R {1} ⊂ {0, 1} {{}} ⊂ {∅, {{∅}}}

Deﬁnition (Proper superset) A set A is a proper superset of set B, denoted A ⊃ B, if and only if B ⊂ A.

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 27 / 56 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?

Basic Deﬁnitions Two or more sets Corollaries of subset deﬁnition

Corollary (Any set is a subset of itself) ∀S ⊆ Ω,S ⊆ S

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56 Corollary ∀A, B such that A ⊂ B, A ⊆ B

Corollary ∀S, ∅ ⊆ S

Can we prove the latter corollary?

Basic Deﬁnitions Two or more sets 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})

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56 Corollary ∀S, ∅ ⊆ S

Can we prove the latter corollary?

Basic Deﬁnitions Two or more sets 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56 Can we prove the latter corollary?

Basic Deﬁnitions Two or more sets 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

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56 Basic Deﬁnitions Two or more sets 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?

Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 28 / 56 T {{1}} ⊆ {1, 2, 3} ? F {{1}} ⊆ {{1}, 2, 3} ? T {{1}} ∈ {{1}, 2, 3} ? F {1} ∈ {{1}, 2, 3} ? T ∅ ⊆ Z ? T ∅ ⊆ { } ? T ∅ ∈ { } ? F ∅ ∈ {{}} ? T ∅ ∈ {{∅}} ? F Master those!