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Sets
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.
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 = {{}}
maka
{c} ∈ R{} ∈ K{} ∉ R
3 ∈ A7 ∉ A{a, b, c} ∈ Rc ∉ R
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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}
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
1 25
3 6
8
4
7A B
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Null Set
Set {{ }} can also be written as {∅}{∅} is not null set because it has one element, that is null set
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
AB
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Subset
Example{ 2, 3, 4} ⊆ {1, 2, 3, 4, 5}{1, 2, 3,44} ⊆ {1, 2, 3, 4}N ⊆ Z ⊆ R ⊆ C
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Subset
TEOREMA 1. For any set AA ⊆ A∅ ⊆ Aif A ⊆ B and B ⊆ C, then A ⊆ C
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
∅ ⊆ 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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
DisjointA and B are called disjoint if both sets do not have similar elementsNotation : A // B
Venn Diagram : U
A B
Example If A = { x | x ∈ Z+, x < 5 } dan B = {20, 30, 40 ... }, then A // B.
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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({∅}) = {∅, {∅}}.
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Set Operations
a. Intersectionb. Unionc. Complementd. Differencee. Symmetric Differencef. Cartesian product
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
IntersectionNotation : A ∩ B = { x | x ∈ A and x ∈ B }
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Union
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 :
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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}
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 }
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Symmetric Difference
TEOREMA
Symmetric Difference meet the following properties:
A ⊕ B = B ⊕ A (commutative law)(A ⊕ B ) ⊕ C = A ⊕ (B ⊕ C )
(associative law)
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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}} }
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Set Identities
Identity laws:A ∪ Ø = AA ∩ U = A
Domination laws:A ∩ Ø = ØA ∪ U = U
Complement laws:A ∪ Ā = UA ∩ Ā = Ø
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Set Identities
Idempotent laws:A ∩ A = AA ∪ A = A
Complementation laws:(Ac)c = A
Absorption laws:A ∪ (A ∩ B) = AA ∩ (A ∪ B) = A
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Set Identities
Commutative laws: A ∪ B = B ∪ AA ∩ B = B ∩ A
associative laws:A ∪ (B ∪ C) = (A ∪ B) ∪ CA ∩ (B ∩ C) = (A ∩ B) ∩ C
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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∩
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Set Identities
0/1 laws
= U= ∅
∅U
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 = ∅
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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⏐
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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.
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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).
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Proving of Statement about Sets
11111111
10111011
11011101
00000001
00001110
00001010
00001100
00000000(A ∩ B) ∪ (A ∩ C)A ∩ CA ∩ BA ∩ (B ∪ C) B ∪ CCBA
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Proving of Statement about Sets
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)
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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)
Proving of Statement about Sets
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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}
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Multisets Operations
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 }
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Multisets Operations
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Fuzzy Sets
A ={ 1, 2, 3, ….,9}
{ }10 than less xxA +Ζ∈=
{ }number big is xxB +Ζ∈=
B=?
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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)
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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∈=
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Fuzzy SetsLet we define degree for several number :
510=x
210=x410=x
5010=x
degree 0
degree 0,3
degree 1
degree 0,35
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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]
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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}
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Fuzzy Sets
Old Teenage
⎪⎪⎩
⎪⎪⎨
⎧
<
≤≤−
>
=
20,0
8020,60
2080,1
)(
x
xxx
xOldµ
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
<<−
<<−
≥≤≤≤
=
3016,14
30
106,4
630atau6,0
1610,1
)(
xx
xxxx
x
xTeenageµ
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 =∪
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 }
Ac =
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 }
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Exercises
Determine whether each of these statements is true or false
φ∈0
{ }0∈φ{ } φ⊂0
{ }0⊂φ
{ } { }00 ⊂
{ } { }00 ⊆
{ } { }φφ ⊆a.
e.
b.
f.
c.
g.
d.
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
Exercises
{ }{ }aa,
{ }{ }a
{ }{ }φφ,
{ } { }{ }{ }aaaa ,,,
{ } { }{ }{ }φφφφ ,,,
a.{a}
b
c
d
e
f
What is the cardinality of each of these sets ?
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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 −
j. k.
Discrete Mathematics (MA 2333) – Faculty of Science Telkom Institute of Technology
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