Chapter I Set - SI-35-02 · Bandung - Indonesia. Discrete Mathematics (MA 2333) – Faculty of...

Discrete Mathematics (MA 2333)Faculty of Science Telkom Institute of TechnologyBandung - Indonesia

Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology

Definition

A set is an unordered collection of objects. The objects in a set are also called the elements, or members, of the set. A set is said to contain its elements.

By enumerating the elementsThe set is described by listing all its members between bracesExamples :

The set of all vowels in the alphabet: V={a,i,u,e,o}The set of odd positive integers less than 10:

O = {13,5,7,9} The set of positive integers less than 100 can be denoted by {1,2,3,...,99}R = { a, b, {a, b, c}, {a, c} }C = {a, {a}, {{a}} }K = { {} }

Set Representations

Set RepresentationsBy enumerating the elements (cont)

MembershipMembership described using notation ∈x ∈ A : x is the element of set A; x ∉ A : x is not the element of set A

ExampleA = {1, 2, 3, 4, 5}, R = { a, b, {a, b, c}, {c} }K = {{}}

{c} ∈ R{} ∈ K{} ∉ R

3 ∈ A7 ∉ A{a, b, c} ∈ Rc ∉ R

Set RepresentationsBy using standard symbols

A set can be expressed using standard symbols that is usually used by academicians

N = the set of natural numbers = { 1, 2, ...}Z = the set of integers ={...,-2, -1, 0, 1, 2,...}Z+ = the set of positive integers = { 1, 2, 3, ...}Q = the set of rational numbersR = the set of real numbersC = the set of complex numbers

Set Representations

By property of the setA set can be described by property of all elements in the set{ x ⎥ property or properties of x }Examples

The set A of positive integers less than 6A = { x | x ∈ Z+, x < 6 } which equivalent with A = {1, 2, 3, 4, 5}

Set RepresentationsBy using Venn Diagram

A set can also represented by listing all its members in a Venn diagramExample

U = {1, 2, …, 7, 8}, A = {1, 2, 3, 5} and

B = {2, 5, 6, 8}.

Venn Diagram :U

CardinalityThe number of elements in a set A is called

the cardinality of the set ADenoted by : n(A) or ⎢A ⎢

ContohB = { x | x is prime numbers less than 10 }, or B = {2, 3, 5, 7} then ⏐B⏐ = 4 A = {a, {a}, {{a}} }, maka ⏐A⏐ = 3

Null SetIf a set does not have any elements, in other words cardinality of the set is zero, the set is called null set

Notation : ∅ or {}

Example E = { x | x < x }, then n(E) = 0A ={x | x is square root of x2 + 1 = 0, and x ∈ R}, n(A)=0

Null Set

Set {{ }} can also be written as {∅}{∅} is not null set because it has one element, that is null set

Subset

The set A is said to be a subset of Bif and only if every element of A is also an element of B. B is said to be superset of A.Notation: A ⊆ BVenn Diagram : U

Subset

Example{ 2, 3, 4} ⊆ {1, 2, 3, 4, 5}{1, 2, 3,44} ⊆ {1, 2, 3, 4}N ⊆ Z ⊆ R ⊆ C

Subset

TEOREMA 1. For any set AA ⊆ A∅ ⊆ Aif A ⊆ B and B ⊆ C, then A ⊆ C

∅ ⊆ A and A ⊆ A, then ∅ and A is called the improper subset of A.

Example :A = {1, 2, 3}, then {1, 2, 3} and ∅ are improper

subset of A.

Statement A ⊆ B is different with A ⊂ B

A ⊂ B : A is subset of B, but A ≠ B.A is proper subset of B.

Example:{1} and {2, 3} are proper subset of {1, 2, 3}

A = B if and only if A ⊆ B and B ⊆ A,

Subset

Equivalent SetsTwo sets are called equivalent if each set has the same cardinality.Notation : A ~ B ↔ ⏐A⏐ = ⏐B⏐

Example Suppose A = { 1, 4, 8, 7 } and B ={ a, b, c, d}, then A ~ B, because ⏐A⏐ = ⏐B⏐ = 4

DisjointA and B are called disjoint if both sets do not have similar elementsNotation : A // B

Venn Diagram : U

Example If A = { x | x ∈ Z+, x < 5 } dan B = {20, 30, 40 ... }, then A // B.

Power SetPower set of A is the set of all subsets of the set A, Notation : P(A) or 2A

If ⏐A⏐ = m, then ⏐P(A)⏐ = 2m.

ExampleIf A = { a, b }, makaP(A) = {∅ , { a }, { b }, { a, b }}P(∅) = {∅} {∅} adalah P({∅}) = {∅, {∅}}.

Set Operations

a. Intersectionb. Unionc. Complementd. Differencee. Symmetric Differencef. Cartesian product

IntersectionNotation : A ∩ B = { x | x ∈ A and x ∈ B }

Intersection

ExamplesIf A = {2, 4, 7, 9, 10} dan B = {4, 10, 9, 18}, maka A ∩ B = {4, 9,10}Jika A = { 3, 5, 7 } dan B = { -4, 8 }, maka

A ∩ B = ∅.

It means: A // B

Notation : A ∪ B = { x | x ∈ A or x ∈ B }

Contoh• If A = { 2, 5, 7 } and B = { 7, 5, 20 }, then

A ∪ B = { 2, 5, 7, 20 }• A ∪ ∅ = A

Complementor Ac = { x | x ∈ U, x ∉ A }A

ExamplesSuppose universe U = { 1, 2, 3, ..., 9 },•If A = {1, 3, 7, 9}, then Ac = {2, 4, 5, 6, 8}•If A = { x | x/2 ∈ P, x < 9 }, then Ac= { 1, 3, 5, 7, 9 }

Notation :

DifferenceNotation : A – B = { x | x ∈ A and x ∉ B } = A ∩ Bc

Examples • if A = { 1, 2, 3, ..., 10 } and B = { 2, 4, 6, 8, 10 }, thenA – B = { 1, 3, 5, 7, 9 } and B – A = ∅

• {1, 3, 5} – {1, 2, 3} = {5}, but {1, 2, 3} – {1, 3, 5} = {2}

Symmetric Difference

Notation: A ⊕ B = (A ∪ B) – (A ∩ B) = (A – B) ∪ (B – A)

Example If A = { 2, 4, 6 } and B = { 2, 3, 5 }, then A ⊕ B = { 3, 4, 5, 6 }

Symmetric Difference

TEOREMA

Symmetric Difference meet the following properties:

A ⊕ B = B ⊕ A (commutative law)(A ⊕ B ) ⊕ C = A ⊕ (B ⊕ C )

(associative law)

Notation: A × B = {(a, b) ⏐ a ∈ A and b ∈ B }

Examples

C = { 1, 2, 3 }, D = { a, b }, then C × D = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }

A = B = set of real numbers, then A × B = set of all dots in a planar shape

CARTESIAN PRODUCT

If A and B are finite sets, then: ⏐A ×B⏐ = ⏐A⏐ . ⏐B⏐Ordered pair (a, b) is different with (b, a), so that (a, b) ≠ (b, a)A × B ≠ B × A, for A or B is not null setIf A = ∅ or B = ∅, then A × B = B × A= ∅

CARTESIAN PRODUCT

CARTESIAN PRODUCTExample : Enumerate all elements of these sets :(a) P(∅) (b) ∅ × P(∅) (c) {∅}× P(∅) (d) P(P({3}))

Answer:(a) P(∅) = {∅}(b) ∅ × P(∅) = ∅(c) {∅}× P(∅) = {∅}× {∅} = {(∅,∅))(d) P(P({4})) = P({ ∅, {4} }) = {∅, {∅}, {{4}}, {∅,

{4}} }

Set Identities

Identity laws:A ∪ Ø = AA ∩ U = A

Domination laws:A ∩ Ø = ØA ∪ U = U

Complement laws:A ∪ Ā = UA ∩ Ā = Ø

Set Identities

Idempotent laws:A ∩ A = AA ∪ A = A

Complementation laws:(Ac)c = A

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

Set Identities

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

associative laws:A ∪ (B ∪ C) = (A ∪ B) ∪ CA ∩ (B ∩ C) = (A ∩ B) ∩ C

Set Identities

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

De Morgan laws:

=∩ BA BA∪

=∪ BA BA∩

Set Identities

0/1 laws

= U= ∅

Duality Principle

Duality Principle: Two different concepts can be exchangeed. The exchange provides true answerExample :

USA → steering is placed in front–leftIndonesia → steering is placed in front–left

Duality PrincipleRules:

In USAcars must take the right side of the roadin the highways, overtaking is done from the left sidewhen the red light is on, cars are allowed to directly turn right

In Indonesiacars must take the left side of the roadin the highways, overtaking is done from the right sidewhen the red light is on, cars are allowed to directly turn left

Duality PrincipleThe left-right principles can be exchanged. Rules applied in USA is used in Indonesia and vice versa

Duality Principle of Set

Suppose that S is identityinvolving set and operations like ∪, ∩ , and complement. If S* is identity that is the dual of S, by changing ∪ → ∩ , ∩ → ∪ , Ø→ U, U →Ø , while the complement is not changed, when the operations are applied to the identity S*, they provide correct answer

Duality Principle of Set

1. Identity law: A ∪ ∅ = A

Dual : A ∩ U = A

2. Domination law: A ∩ ∅ = ∅

Dual : A ∪ U = U

3. Complement law:

A ∪ A = U

Dual: A ∩ A = ∅

4. Idempotent law:

A ∪ A = A

Dual : A ∩ A = A

5. Absorption law :

A ∪ (A ∩ B) = A

Dual: A ∩ (A ∪ B) = A

Duality Principle of Set6. Commutative law:

A ∪ B = B ∪ A

Dual : A ∩ B = B ∩ A

7. Associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C

Dual: A ∩ (B ∩ C) = (A ∩ B) ∩ C

8. Distributive law: A ∪ (B ∩ C)=(A ∪ B) ∩ (A ∪ C)

Dual: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

9. De Morgan law: BA∪ = A ∩ B

Dual : BA∩ = A ∪ B

10. 0/1 law

∅= U

Dual :

U = ∅

Inclusion-Exclusion Principle

Given two sets A and B:⏐A ∪ B⏐ = ⏐A⏐ + ⏐B⏐ – ⏐A ∩ B⏐⏐A ⊕ B⏐ = ⏐A⏐ +⏐B⏐ – 2⏐A ∩ B⏐

Given tree sets A, B, and C :⏐A ∪ B ∪ C⏐ = ⏐A⏐ + ⏐B⏐ + ⏐C⏐ – ⏐A ∩ B⏐ – A ∩C⏐– ⏐B ∩ C⏐ + ⏐A ∩ B ∩ C⏐

Inclusion-Exclusion Principle

Example: How many integers not exceeding 1000 are divisible by 7 or 11Solution: ⏐A⏐ = ⎣1000/7⎦ = 142, ⏐B⏐ = ⎣1000/11⎦ = 90, ⏐A ∩ B⏐ = ⎣1000/77⎦ = 12⏐A ∪ B⏐ = ⏐A⏐ + ⏐B⏐ – ⏐A ∩ B⏐

= 142 + 90 – 12 = 220

There are 47 integers that divisible by either 7 or 11.

Proving of Statement about Sets

Proving by membership table

Example:Let A, B, and C be sets. Proof that A ∩(B ∪ C) = (A ∩ B) ∪ (A ∩ C).

11111111

10111011

11011101

00000001

00001110

00001010

00001100

00000000(A ∩ B) ∪ (A ∩ C)A ∩ CA ∩ BA ∩ (B ∪ C) B ∪ CCBA

Proving by Venn diagramExampleLet A, B, and C be setsProof that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) by Venn Diagram

Solution:

A ∩ (B ∪ C) (A ∩ B) ∪ (A ∩ C)

Proving by set algebraGiven sets A and BProof that (A ∩ B) ∪ (A ∩ Bc) = ASolution: (A ∩ B) ∪ (A ∩ Bc) = A ∩ (B ∪ Bc) (Distributive law)

= A ∩ U (Complement law)= A (Identity law)

MultisetsMultisets are unordered collections of elements where an element can occur as a member more than once examples : A = {1, 1, 2, 2, 2, 3}, B = {3, 3, 3}, C = {2, 3, 4}, D = {}. Multiplicities of an element in a multiset is the number of occurrence of the element in the multiset. Examples: M = { 0, 2, 2, 2, 0, 0, 0, 1 }, multiplicity of 0 is 4, multiplicity of 1 is 1, and multiplicity of 2 is 3

Multisets

Set is a special case of multiset. In this case, the multiplicities of its element are 1 or 0.A set which multiplicities of its elements are 0 is null set

Multisets Operations

Let P and Q be multisets:P U Q is the multiset where the multiplicity of an element is the maximum of its multiplicities in P and Q.

Example: P = {a,a,a,c,d,d} and Q ={a,a,b,c,c}, P ∪ Q = {a,a,a,b,c,c,d,d }P ∩ Q is the multiset where the multiplicity of an element is the minimum of its multiplicities in P and Q.

Example: P = {a,a,a,c,d,d} dan Q = {a,a,b,c,c}

P ∩ Q = {a,a,c}

P – Q is the multiset where the multiplicity of an element is the multiplicity of the element in P less its multiplicity in Q unless this difference is negative, in which case the multiplicity is 0Example: P = { a, a, a, b, b, c, d, d, e } Q = { a, a, b, b, b, c, c, d, d, f } P – Q = { a, e }

P + Q, is the multiset where the multiplicity of an element is the sum of multiplicities in P and QExample: P = { a, a, b, c, c } Q = { a, b, b, d },P + Q = { a, a, a, b, b, b, c, c, d }

Fuzzy Sets

Classical set theory requires that a set must have a well-defined property In practical, some sets do not have clear boundaries → fuzzy set theory

Fuzzy Sets

A ={ 1, 2, 3, ….,9}

{ }10 than less xxA +Ζ∈=

{ }number big is xxB +Ζ∈=

Fuzzy Sets

Fuzzy sets are used in artificial intelligence. Each element in the universal set U has a degree of membership, which is a real number in interval [0,1]In enumeration representation, The fuzzy set denoted by listing the elements with their degrees of membership (element with 0 degree of membership are not listed)

Fuzzy SetsLet we define a set :

”Big number” can be defined very relatively. For example, 10,000 can be said as ”big” or ”not big” number. Therefore, we need degree to represent how big 10,000 is. If we define membership degree of 10,000 is as much as 0.3, we can also define the degree of other natural number.

{ }number natural big is xxB +Ζ∈=

Fuzzy SetsLet we define degree for several number :

210=x410=x

5010=x

degree 0

degree 0,3

degree 1

degree 0,35

Fuzzy SetsA fuzzy set in a universe of discourse U is characterized by a membership function )(xAµthat takes values in the interval [0,1]

A fuzzy set is generalization of classical set by allowing the membership function to take any values in the interval [0,1]In other words, the membership function of a classical set can only take two values-zero and one, whereas the membership function of a fuzzy set is a continuous function with range [0,1]

Fuzzy SetsA fuzzy set A in U may be represented as a set of ordered pairs of a generic element x and its membership value, that is,

( )( ){ }UxxxA A ∈= µ,

ExampleLet F be a fuzzy set named “natural numbers close to six”. Then a possible membership function for F is

={0,1/3 + 0,3/4 + 0,6/5 + 1,0/6 + 0,6/7 + 0,3/8 + 0,1/9}

)(xFµU = set of natural numbers

Where x ∈ U

Fuzzy SetsExample

Let U be the integers from 1 to 10, that is U = {1,2,3,...,10}. Then the fuzzy set “several”may be defined asB = set of “several”

)(xFµ = {0,5/3 + 0,8/4 + 1/5 + 1/6 + 0,8/7 + 0,5/8}

Fuzzy Sets

How to determine the membership functions?

Using the knowledge of human expertscan only give a rough formula of the membership function – fine tuning is requiredUsing data collected from various sensors to determine the membership functions

Fuzzy SetsWhen U is continuous, a fuzzy set A is commonly written as

( )∫=U

A xxA /µ

where the integral sign does not denote integration; it denotes the collection of all points x ∈ U with the associated membership function )(xAµ

Fuzzy Sets

Example Let U be interval [0,100] representing the age of ordinary humans. Then we may define fuzzy sets ”old”and ”teenage” as

Fuzzy Sets

Old Teenage

⎪⎪⎩

⎪⎪⎨

≤≤−

8020,60

2080,1

xOldµ

⎪⎪⎪

≥≤≤≤

3016,14

630atau6,0

1610,1

xTeenageµ

Fuzzy Sets OperationsWe assume that A and B are fuzzy set which defined on the same universe

)(1)( xx AAc µµ −=Complement

Intersection ( ))(),(min )( xxx BABA µµµ =∩

Union ( ))(),(max )( xxx BABA µµµ =∪

Fuzzy Sets Operations

U = {1, 2, 3, 4, 5, 6}A = { 0/1 + 0,2/2 + 0,6/3 + 0,9/4 + 1/5 + 0,8/6 }B = { 0,8/1 + 1/2 + 0,7/3 + 0,4/4 + 0,1/5 + 0/6 }

)(1)( xx AAc µµ −=

= { 1/1 + 0,8/2 + 0,4/3 + 0,1/4 + 0/5 + 0,2/6 }

BA∩ ( ))(),(min )( xxx BABA µµµ == ∩

= { 0/1 + 0,2/2 + 0,6/3 + 0,4/4 + 0,1/5 + 0/6 } BA∪ ( ))(),(max )( xxx BABA µµµ == ∪

= { 0,8/1 + 1/2 + 0,7/3 + 0,9/4 + 1/5 + 0,8/6 }

Exercises

Determine whether each of these statements is true or false

φ∈0

{ }0∈φ{ } φ⊂0

{ }0⊂φ

{ } { }00 ⊂

{ } { }00 ⊆

{ } { }φφ ⊆a.

Exercises

{ }{ }aa,

{ }{ }a

{ }{ }φφ,

{ } { }{ }{ }aaaa ,,,

{ } { }{ }{ }φφφφ ,,,

What is the cardinality of each of these sets ?

Exercises

Given setsU = {1, 2, 3, …., 8, 9} → UniverseA = {1, 2, 3, 4} B = {2, 4, 6, 8}C = {3, 4, 5, 6}

BA∪CA∪CB∪

BB ∪

a. b. c. d.

( ) CBA ∪∪( )CBA ∪∪( )CBA ∪∩

( ) ( )CABA ∩∪∩

e. f. g. h.

( )cBA∪cc BA ∩

h. i. ( ) CBA −∩( )cBA −

References

Rosen, Kenneth H., Discrete Mathematics and Its Applications 5th Ed, McGraw-Hill, New York, 2003Munir, Rinaldi., Matematika Diskrit, Edisi Kedua, Penerbit InformatikaBandung, Bandung, 2003

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