Discrete Mathematics and Its Applicationsdase.ecnu.edu.cn/mgao/teaching/DM_2020_Spring/slides/2_sets22.pdf · Given a set S, the power set of S is the set of all subsets of S, denoted

Discrete Mathematics and Its ApplicationsLecture 2: Basic Structures: Set Theory

MING GAO

DaSE@ ECNU(for course related communications)

[email protected]

Mar. 27, 2020

Outline

1 Set Concepts

2 Set Operations

3 Application

4 Take-aways

MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 2 / 16

Set Concepts

Set

Definition

Set A is a collection of objects (or elements).

a ∈ A : “a is an element of A” or “a is a member of A”;

a 6∈ A : “a is not an element of A”;

A = {a1, a2, · · · , an} : A contains a1, a2, · · · , an;

Order of elements is meaningless;

It does not matter how often the same element is listed.

Set equality

Sets A and B are equal if and only if they contain exactly the same elements.

If A = {9, 2, 7,−3} and B = {7, 9, 2,−3}, then A = B.

If A = {9, 2, 7} and B = {7, 9, 2,−3}, then A 6= B.;

If A = {9, 2,−3, 9, 7,−3} and B = {7, 9, 2,−3}, then A = B.


Set Concepts

Set

Definition

Set A is a collection of objects (or elements).

a ∈ A : “a is an element of A” or “a is a member of A”;

a 6∈ A : “a is not an element of A”;

A = {a1, a2, · · · , an} : A contains a1, a2, · · · , an;

Order of elements is meaningless;

It does not matter how often the same element is listed.

Set equality

Sets A and B are equal if and only if they contain exactly the same elements.

If A = {9, 2, 7,−3} and B = {7, 9, 2,−3}, then A = B.

If A = {9, 2, 7} and B = {7, 9, 2,−3}, then A 6= B.;

If A = {9, 2,−3, 9, 7,−3} and B = {7, 9, 2,−3}, then A = B.MING GAO (DaSE@ecnu) Discrete Mathematics and Its Applications Mar. 27, 2020 3 / 16

Set Concepts

Applications

Examples

Bag of words model: documents, reviews, tweets, news, etc;

Transactions: shopping list, app downloading, book reading,video watching, music listening, etc;

Records in a DB, data item in a data streaming, etc;

Neighbors of a vertex in a graph;

“Standard” sets

Natural numbers: N = {0, 1, 2, 3, · · · }Integers: Z = {· · · ,−2,−1, 0, 1, 2, · · · }Positive integers: Z+ = {1, 2, 3, 4, · · · }Real Numbers: R = {47.3,−12,−0.3, · · · }Rational Numbers: Q = {1.5, 2.6,−3.8, 15, · · · }


Set Concepts

Applications

Examples

Bag of words model: documents, reviews, tweets, news, etc;

Transactions: shopping list, app downloading, book reading,video watching, music listening, etc;

Records in a DB, data item in a data streaming, etc;

Neighbors of a vertex in a graph;

“Standard” sets

Natural numbers: N = {0, 1, 2, 3, · · · }Integers: Z = {· · · ,−2,−1, 0, 1, 2, · · · }Positive integers: Z+ = {1, 2, 3, 4, · · · }Real Numbers: R = {47.3,−12,−0.3, · · · }Rational Numbers: Q = {1.5, 2.6,−3.8, 15, · · · }


Set Concepts

Representation of sets

Tabular form

A = {1, 2, 3, 4, 5};B = {−2, 0, 2};

Descriptive form

A = set of first five natural numbers;

B = set of positive odd integers;

Set builder form

Q = {a/b : a ∈ Z ∧ b ∈ Z ∧ b 6= 0};B = {y : P(y)}, where P(Y ) : y ∈ E ∧ 0 < y ≤ 50;


Set Concepts


Tabular form

A = {1, 2, 3, 4, 5};B = {−2, 0, 2};

Descriptive form



Set builder form



Set Concepts


Tabular form

A = {1, 2, 3, 4, 5};B = {−2, 0, 2};

Descriptive form



Set builder form



Set Concepts


Remarks

A = ∅ : empty set, or null set;

Universal set U: contains all the objects under consideration.

A = {{a, b}, {b, c , d}};

Venn diagrams

In general, a universal set is represented by a rectangle.


Set Concepts


Remarks

A = ∅ : empty set, or null set;

Universal set U: contains all the objects under consideration.

A = {{a, b}, {b, c , d}};

Venn diagrams

In general, a universal set is represented by a rectangle.


Set Concepts

Subsets

Definition

Set A is a subset of B iff every element of A is also an element of B,denoted as A ⊆ B.

A ⊆ B = ∀x(x ∈ A→ x ∈ B);

For every set S , we have: (1) ∅ ⊆ S ; and (2) S ⊆ S ;

When we wish to emphasize that set A is a subset of set B butthat A 6= B, we write A ⊂ B and say that A is a proper subsetof B, i.e., ∀x(x ∈ A→ x ∈ B) ∧ ∃(x ∈ B ∧ x 6∈ A);

Two useful rules: (1) A = B ⇔ (A ⊆ B) ∧ (B ⊆ A);(2) (A ⊆ B) ∧ (B ⊆ C )⇒ (A ⊆ C );

Given a set S , the power set of S is the set of all subsets of S ,denoted as P(S). The size of 2|S |, where |S | is the size of S .


Set Concepts

Subsets

Definition


A ⊆ B = ∀x(x ∈ A→ x ∈ B);






Set Concepts

Subsets

Definition


A ⊆ B = ∀x(x ∈ A→ x ∈ B);






Set Concepts

Subsets

Definition


A ⊆ B = ∀x(x ∈ A→ x ∈ B);






Set Concepts

Subsets

Definition


A ⊆ B = ∀x(x ∈ A→ x ∈ B);






Set Concepts

Subsets

Definition


A ⊆ B = ∀x(x ∈ A→ x ∈ B);






Set Concepts

Subsets

Definition


A ⊆ B = ∀x(x ∈ A→ x ∈ B);






Set Concepts

Cartesian product

Definition

Let A and B be sets. The Cartesian product of A and B, denotedby A× B, is set A× B = {(a, b) : a ∈ A ∧ b ∈ B}, where (a, b) is aordered 2-tuples, called ordered pairs.

Let A = {1, 2} and B = {a, b, c}, the Cartesian product

A× B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)};

The Cartesian product of A1,A2, · · · ,An is denoted asA1 × A2 × · · · × An

A1 × A2 × · · · × An = {(a1, a2, · · · , an) : ∀i ai ∈ Ai};

A subset R of the Cartesian product A× B is called a relationfrom A to B.


Set Concepts

Cartesian product

Definition



A× B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)};


A1 × A2 × · · · × An = {(a1, a2, · · · , an) : ∀i ai ∈ Ai};



Set Concepts

Cartesian product

Definition



A× B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)};


A1 × A2 × · · · × An = {(a1, a2, · · · , an) : ∀i ai ∈ Ai};



Set Concepts

Cartesian product

Definition



A× B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)};


A1 × A2 × · · · × An = {(a1, a2, · · · , an) : ∀i ai ∈ Ai};



Set Operations

Set operations

Operators

Let A and B be two sets, and U be the universal set

Union: A ∪ B = {x : x ∈ A ∨ x ∈ B};Intersection: A ∩ B = {x : x ∈ A ∧ x ∈ B};Difference: A− B = {x : x ∈ A ∧ x 6∈ B} (sometimes denotedas A \ B);

Complement: A = U − A = {x ∈ U : x 6∈ A};

A− B = A ∩ B;

|A ∪ B| = |A|+ |B| − |A ∩ B|.


Set Operations

Set operations

Operators




A− B = A ∩ B;

|A ∪ B| = |A|+ |B| − |A ∩ B|.


Set Operations

Set operations

Operators




A− B = A ∩ B;

|A ∪ B| = |A|+ |B| − |A ∩ B|.


Set Operations

Set operations

Operators




A− B = A ∩ B;

|A ∪ B| = |A|+ |B| − |A ∩ B|.


Set Operations

Set identities

Table of logical equivalence

equivalence name

A ∩ U = A Identity lawsA ∪ ∅ = A

A ∩ A = A Idempotent lawsA ∪ A = A

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

A ∪ (A ∩ B) = A Absorption lawsA ∩ (A ∪ B) = A

A ∩ A = ∅ Complement laws

A ∪ A = U


Set Operations

Logical equivalence Cont’d

Table of logical equivalence

equivalence name

A ∪ U = U Domination lawsA ∩ ∅ = ∅

A ∩ B = B ∩ A Commutative lawsA ∪ B = B ∪ A

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

(A ∩ B) = A ∪ B De Morgan’s laws

(A ∪ B) = A ∩ B


Set Operations

Proof of De Morgan law

Proof

Use set builder notation and logical equivalences to establish the firstDe Morgan lawSteps Reasons

A ∩ B primise

= {x : x 6∈ A ∩ B} definition of complement= {x : ¬(x ∈ A ∩ B)} definition of does not belong symbol= {x : ¬(x ∈ A ∧ x ∈ B)} definition of intersection= {x : ¬(x ∈ A) ∨ ¬(x ∈ B)} De Morgan law for logic= {x : (x 6∈ A) ∨ (x 6∈ B)} definition of does not belong symbol

= {x : (x ∈ A) ∨ (x ∈ B)} definition of complement

= {x : x ∈ (A ∪ B)} definition of union

= A ∪ B meaning of set builder notation


Set Operations


Proof


A ∩ B primise= {x : x 6∈ A ∩ B} definition of complement

= {x : ¬(x ∈ A ∩ B)} definition of does not belong symbol= {x : ¬(x ∈ A ∧ x ∈ B)} definition of intersection= {x : ¬(x ∈ A) ∨ ¬(x ∈ B)} De Morgan law for logic= {x : (x 6∈ A) ∨ (x 6∈ B)} definition of does not belong symbol





Set Operations


Proof


A ∩ B primise= {x : x 6∈ A ∩ B} definition of complement= {x : ¬(x ∈ A ∩ B)} definition of does not belong symbol

= {x : ¬(x ∈ A ∧ x ∈ B)} definition of intersection= {x : ¬(x ∈ A) ∨ ¬(x ∈ B)} De Morgan law for logic= {x : (x 6∈ A) ∨ (x 6∈ B)} definition of does not belong symbol





Set Operations


Proof


A ∩ B primise= {x : x 6∈ A ∩ B} definition of complement= {x : ¬(x ∈ A ∩ B)} definition of does not belong symbol= {x : ¬(x ∈ A ∧ x ∈ B)} definition of intersection

= {x : ¬(x ∈ A) ∨ ¬(x ∈ B)} De Morgan law for logic= {x : (x 6∈ A) ∨ (x 6∈ B)} definition of does not belong symbol





Set Operations


Proof


A ∩ B primise= {x : x 6∈ A ∩ B} definition of complement= {x : ¬(x ∈ A ∩ B)} definition of does not belong symbol= {x : ¬(x ∈ A ∧ x ∈ B)} definition of intersection= {x : ¬(x ∈ A) ∨ ¬(x ∈ B)} De Morgan law for logic

= {x : (x 6∈ A) ∨ (x 6∈ B)} definition of does not belong symbol





Set Operations


Proof


A ∩ B primise= {x : x 6∈ A ∩ B} definition of complement= {x : ¬(x ∈ A ∩ B)} definition of does not belong symbol= {x : ¬(x ∈ A ∧ x ∈ B)} definition of intersection= {x : ¬(x ∈ A) ∨ ¬(x ∈ B)} De Morgan law for logic= {x : (x 6∈ A) ∨ (x 6∈ B)} definition of does not belong symbol





Set Operations


Proof







Set Operations


Proof







Set Operations


Proof







Set Operations

Generalized unions and intersections

Union

A1 ∪ A2 ∪ · · · ∪ An =⋃n

i=1 Ai ;

A1 ∪ A2 ∪ · · · ∪ An ∪ · · · =⋃∞

i=1 Ai ;

Intersection

A1 ∩ A2 ∩ · · · ∩ An =⋂n

i=1 Ai ;

A1 ∩ A2 ∩ · · · ∩ An ∩ · · · =⋂∞

i=1 Ai ;


Set Operations

Generalized unions and intersections

Union

A1 ∪ A2 ∪ · · · ∪ An =⋃n

i=1 Ai ;

A1 ∪ A2 ∪ · · · ∪ An ∪ · · · =⋃∞

i=1 Ai ;

Intersection

A1 ∩ A2 ∩ · · · ∩ An =⋂n

i=1 Ai ;

A1 ∩ A2 ∩ · · · ∩ An ∩ · · · =⋂∞

i=1 Ai ;


Application

Set covering problem

Input

Universal set U = {u1, u2, · · · , un}Subsets S1,S2, · · · , Sm ⊆ UCost c1, c2, · · · , cm

Goal

Find a set I ⊆ {1, 2, · · · ,m} that minimizes∑

i∈I ci ,such that

⋃i∈I Si = U

Applications

Document summarization

Natural language generation

Information cascade


Application


Input


Goal


i∈I ci ,such that

⋃i∈I Si = U

Applications



Information cascade


Application


Input


Goal


i∈I ci ,such that

⋃i∈I Si = U

Applications



Information cascade


Take-aways

Take-aways

Set Concepts;

Set Operations;

Applications.


Documents

Discrete Mathematics and Its Applicationsdase.ecnu.edu.cn/mgao/teaching/DM_2020_Spring/slides/2_sets22.pdf · Given a set S, the power set of S is the set of all subsets of S, denoted