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Introduction (Pengenalan) About the Lecturer:– Nama lengkap: Heru Suhartanto, Ph.D– Kantor: Ruang 1214, Gedung A, Fakultas Ilmu Komputer UI, Depok– E-mail: heru@cs.ui.ac.id– Pendidikan formal:

– Sarjana Matematika UI, 1986– Master of Science, Computer Science, University of Toronto, Canada, 1990.– Philosiphy Doctor (Ph.D), Parallel Computing, University of Queensland,

Australia, 1998. Other lecturers

– Achmad Nizar Hidayanto– Ade Azurat– Kasiyah M. Yunus– Dina Cahyati– Siti Aminah

Materi Matrikulasi Matematika – pengenalan (lihat Outline), sebagian diberikan dalam text bahasa Inggris.

Materi: http://telaga.cs.ui.ac.id/WebKuliah/Matrikulasi/math/

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Lecture 1

Set Theory

Reading: Chp 5

Susanna S. Epp, Discrete Mathematics with Application 2-nd Ed, Brooks/Cole, 1995

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1. Sets

1.1 (Definition: Set)

A SET is an unordered collection of unique elements.

Notation: It is written as:

{x1,…,xn}

where n 0 and x1,…,xn are the elements of the set.

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1. Sets

1.2 Examples of sets– {1, 24, 32}– {apple, car, pencil}– {,,,}– {1, apple, }– {{1,2}, apple, { {},{,3}}}– {} is a set with no elements. It is known as

the empty set and is also denoted as ‘’

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1. Sets

1.3 Remarksa. Ordering does not matter.

{1,2,3} = {1,3,2} = {2,1,3}

b. Repetitions are ignored.

{1,1,2,3} = {1,2,3}

c. Elements in the set need not be of the ‘same type’.

{1, apple, } is a set

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1. Sets

1.3 Remarks (cont’d)d. A set can contain other sets as

elements

{{1,2}, apple, {{},{,3}}}

is a set with 3 elements:• {1,2}• apple• {{},{,3}}

e. A set can be finite or infinite.

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1. Sets

1.4 Predefined Sets– The set of Natural numbers

N = {0, 1, 2, 3,…}– The set of Integers

Z = {…,-2,-1,0,1,2,…}– The set of Rational numbers

Q = {a/b | aZ bZ b0}– The set of Real numbers: R

Real numbers comprise all rational (eg. 1/2) and all irrational numbers (eg. 2).

(Note: There are numbers which are not real numbers, these are not covered in this course).

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1. Sets

1.4 Predefined Sets (cont’d)– The superscript ‘+’ to Z, Q or R indicates

positive numbers (> 0)– The superscript ‘–’ to Z, Q or R indicates

negative numbers (< 0)– The superscript ‘nonneg’ to Z, Q or R

indicates positive numbers including 0.– Therefore, given that Z = {…,-2,-1,0,1,2,…},

Z+ = {1,2,3,…}

Z- = {-1,-2,-3,…}

Znonneg = {0,1,2,3,…}

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1. Sets

1.5 Defining a Set– A set may be defined directly by listing every

element:

S = {2, 4, 6, 8, 10}– Or it may be defined indirectly by defining it in

terms of other sets:

S = {x | x Z, 1 x 10}

S = {x Z | 1 x 10}

Note: Read the symbol ‘|’ as ‘such that’– In general,

S = {element | element Another set, list of conditions}

S = {element Another set | list of conditions}

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2. Visualization tool: Venn Diagram

A Venn Diagram is used to visualize relationships between sets.

1. Draw Sets as Circles. – Spatial relationship between circles is used to

depict set relationships

2. Draw Elements as Dots.

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Outline

Sets– Defn & Notation– Examples– Remarks– Predefined Sets– Defining a set

Venn Diagrams Predicates

– Membership ()– Subset ()– Equality ()– Proper Subset ()

Functors– Union ()– Intersection ()– Difference ()– Complement (c)

Proofs Special sets

– Empty Set– Universal Set– Proofs

Set Equivalences More operations on sets

– Power Set– Cartesian product– Disjoint Unions

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3. Predicates: 3.1 Definition: Set Membership ()

– If x is an element of a set A, we write

x AWe say “x is in A”, “x is a member of A”, or “x is an element of A”

– If x is NOT an element of a set A, we write x A

which is actually an abbreviation of(x A)

A2 1

1 A, 2 A

Venn Diagram:

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3. Predicates:

3.1.1 Examples of ‘’:

• 1 {1, 2, 3}

• 1 {{1,2}, {4}, 5}

• {1} {{1,2}, {4}, 5}

• {1,2} {{1,2}, {4}, 5}

• {1,2} {1, 2, 3, 4, 5}

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3. Predicates:

3.2 Definition: Subset (). Given 2 sets A and B,

A B iff x, xA xB

A

A B

B

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3. Predicates:

Examples

3.2 Definition: Subset (). Given 2 sets A and B,

A B iff x, xA xB

• {1,2} {{1,2}}• {1,2} not {1,{2}}

• {1,2} {1,2,3}

• {1,2} Z

• {} {1,2}

• Is 2 {1,2,3} ?• Is {2} {1,2,3} ?

• Is {2} {2,{2}} ?

• Is 2 {1,2,3} ?• Is {2} {1,2,3} ?

• Is {2} {2,{2}} ?

Note the difference between ‘’ and ‘’. No.

Yes.

Yes.

Yes.

No.

Yes.

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3. Predicates:

3.3 Definition: Set Equality (). Given 2 sets A, B,

A B iff A B B A

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3. Predicates:

3.4 Definition: Proper Subset (). Given 2 sets A and B,

A B iff A B A B

A

A B

B

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3. Predicates:

3.4 Definition: Proper Subset (). Given 2 sets A and B,

A B iff A B A B

Example:– {1,2} {1,2}– {1,2} {1,2,3}– Z+ Z– Z Q– Q R

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Outline

Sets– Defn & Notation– Examples– Remarks– Predefined Sets– Defining a set

Venn Diagrams Predicates

– Membership ()– Subset ()– Equality ()– Proper Subset ()

Functors (Operation)– Union ()– Intersection ()– Difference ()– Complement (c)

Proofs Special sets

– Empty Set– Universal Set– Proofs

Set Equivalences More operations on sets

– Power Set– Cartesian product– Disjoint Unions

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4. Operations (Functors) on sets

If A and B are sets, then

(a) A B (set union)

(b) A B (set intersection)

(c) A B (set difference)

(d) Ac (set complement)

are sets that obey the following axiomatic definitions:

– x, x (A B) iff xA xB– x, x (A B) iff xA xB– x, x (A B) iff xA xB– x, x Ac iff xA

Daffy-nitions

Don’t leave home without them!!!

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4. Operations (Functors) on sets

A B

A B

BA

A B

BA

A B

A

Ac

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5. Proofs

5.2 Prove that A (B C) (A B) (A C)

Proof:

Assume e A (B C) e A e (B C) e A (e B e C)

A (B C) (A B) (A C)

(e A e B) (e A e C) (e A B) (e A C)

e (A B) (A C)

(A B) (A C) A (B C)

Therefore A (B C) (A B) (A C)

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5. Proofs

5.3 Prove that (A B)c Ac Bc

Proof:

Assume e (A B)c

e (A B)

(A B)c Ac Bc

~(e (A B)) ~(e A e B) e A e B

Ac Bc (A B)c

Therefore (A B)c Ac Bc

e Ac e Bc

e Ac Bc

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5. Proofs

5.4 Prove that if A B then A B B

Proof:

e A e B

Case 1: e A

e B (Since A B)

Case 2: e B

e A e BAssume e A B

Therefore, if A B then A B B

e B

Assume e B

e A B

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Outline

Sets– Defn & Notation– Examples– Remarks– Predefined Sets– Defining a set

Venn Diagrams Predicates

– Membership ()– Subset ()– Equality ()– Proper Subset ()

Functors– Union ()– Intersection ()– Difference ()– Complement (c)

Proofs Special sets

– Empty Set– Universal Set– Proofs

Set Equivalences More operations on sets

– Power Set– Cartesian product– Disjoint Unions

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6.1 The Empty Set

Definition: The Empty Set ()– {} is a set with NO elements. – It is known as the empty set and is also denoted

as – It obeys the following axiom:

x, x {}or, worded in another way:

(x, x A) A = {} Misconceptions About the Empty Set:

– {} is an empty set– {{}} is NOT an empty set.

• {{}} has one element: {}• Always look at the outer brackets

– {{},{{}}} is NOT an empty set.

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6.2 The Universal Set

Definition: The Universal Set (U)– U is a set with ALL elements. – It is known as the universal set– It obeys the following axiom:

x, x U

or, worded in another way:

(x, x A) A = U

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6.3 Proofs involving and U

6.3.1 Theorem: For any set A, A.

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6.3 Proofs involving and U

6.3.2 Show that there is only one empty set.

Q: How do we express the idea of ‘only one’?

A: Express it indirectly: ‘there cannot be two’

x, y, If P(x) and P(y), then x = y

Proof: if 1 and 2 be 2 empty sets, then 1 2 .

– Let 1 and 2 be 2 empty sets.

– By previous theorem, 1 2

(Since the empty set 1 must be the subset of any set)

– Also by previous theorem, 2 1

(Since the empty set 2 must be the subset of any set)

– Therefore 1 2, (by definition of set equality).

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6.3 Proofs involving and U

6.3.3 Show that A A (Identity Law)

Proof:

Assume e A e A e e (since axiom of empty set: x, x )

e A

Assume e A e A e e A

Note that you can’t go backwards. As long as there is one reason used in the forward direction which is not an IFF reason, the way back is broken.

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6.3 Proofs involving and U

6.3.4 Show that A (Universal Bound Law)

Proof:

e A e A e

BUT e (Since x, x )

We just need to show that A has no elements.

Remember the axiom: (x, x ???) ??? = {}

e

(By contradiction): Assume A has some element e.

Contradiction!Therefore e A .

Therefore A has no elements.

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6.3 Proofs involving and U

6.3.5 Show that A Ac U (Complementation Law)

Proof:

e U

e A e A

e A e Ac

e A Ac

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7. Set Equivalences

Set Equivalences are very similar to Logical Equivalences

– Intersection similar to – Union similar to – Complement similar to ~– Universal set similar to T– Empty set () similar to

List of identities in p247 and p260 of textbook

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Outline

Sets– Defn & Notation– Examples– Remarks– Predefined Sets– Defining a set

Venn Diagrams Predicates

– Membership ()– Subset ()– Equality ()– Proper Subset ()

Functors– Union ()– Intersection ()– Difference ()– Complement (c)

Proofs Special sets

– Empty Set– Universal Set– Proofs

Set Equivalences More operations on sets

– Power Set– Cartesian product– Disjoint Unions

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8 Power Set

8.1 Definition (Power Set): – Given a set A, the power set of A, denoted as P(A)

is the set of all subsets of A.– It obeys the following axiom:

S, (S A) (S P(A))

Examples:– A = {1,2}, P(A) = {{},{1},{2},{1,2}}– A = {1,2,3}, P(A)={{},{1},{2},{3},{1,2},{1,3},{2,3},

{1,2,3}}– A = {{1},{{2}}}

P(A)={{},{{1}},{{{2}}},{{1},{{2}}}}

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8 Power Set , exercises

8.2 Show that for all sets: if A B, then P(A) P(B)

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8 Power Set

Theorem: If A has n elements,

then P(A) has 2n elements.

Proof in recommended text (p264,p265)

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9 Ordered n-tuple

9.1 Definition: (Ordered n-tuple)

Let n be a positive integer and x1,…,xn be (not necessarily unique) elements. An ordered n-tuple is a collection of n objects denoted as:

(x1,…,xn)

with x1 as the first element, x2 as the second element…xn as the nth element.

NOTE: Ordering of elements is important!

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9 Ordered n-tuple

9.2 Examples:– (1,4,2,5,2) is an ordered 5-tuple– (4,3,3,4) is an ordered 4-tuple– (1,3,1) is an ordered 3-tuple, also known as

an ordered triplet.– (5,3) is an ordered 2-tuple, also known as

an ordered pair.– (3) is an ordered 1-tuple, also known as an

singleton.

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9 Ordered tuples

9.3 Definition (Equality of ordered tuples) (x1,…,xn) = (y1,…,ym) iff

n=m and x1 = y1 and x2=y2 and … and xn=yn

9.4 Examples:– (1,a) (1,a,c)– (1,a,c) (1,c,a)– (1,a,c) (1,a,c)– (2,4,3) (1+1,22,5-2)

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10. Cartesian Product

10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian

product of A and B is denoted as A x B.

– It obeys the following axiom:(x,y) A B iff xA yB

– We can also write:A B = { (x,y) | xA yB}

Examples:– {1,2} x {2,3} = {(1,2),(1,3),(2,2),(2,3)}

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10. Cartesian Product

10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian

product of A and B is denoted as A x B.

– It obeys the following axiom:(x,y) A B iff xA yB

– We can also write:A B = { (x,y) | xA yB}

Examples:– {1,2,3} x {a,b}

= {(1,a),(2,a),(3,a), (1,b),(2,b),(3,b)}

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10. Cartesian Product

10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian

product of A and B is denoted as A x B.

– It obeys the following axiom:(x,y) A B iff xA yB

– We can also write:A B = { (x,y) | xA yB}

Examples:– {{1},2,{3,4}} x {a,b}

= { ({1},a), (2,a), ({3,4},a),

({1},b), (2,b), ({3,4},b)}

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10. Cartesian Product

10.1 Definition (Cartesian Product)– Given 2 Sets A and B, the cartesian

product of A and B is denoted as A x B.

– It obeys the following axiom:(x,y) A B iff xA yB

– We can also write:A B = { (x,y) | xA yB}

Q: {1,2} x {} = ? A: {}

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10. Cartesian Product

10.2 Definition (Generalised definition of cartesian product):

Given sets A1,…,An, A1 A2 … An is the set of all ordered n-tuples (x1,…,xn) where x1A1 x2A2 … xnAn

Examples:{1,2} x {2,3} x {a,b}

= {(1,2,a), (1,2,b), (1,3,a), (1,3,b), (2,2,a),

(2,2,b), (2,3,a), (2,3,b)}

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10. Cartesian Product (Proofs)

10.3 Show that A x (B C) (A x B) (A x C)

Proof:

Assume (m,n) A x (B C) m A n (B C) m A (n B n C) (m A n B) (m A n C) ((m,n) A x B) ((m,n) A x C)

(m,n) (A x B) (A x C)

Therefore A x (B C) (A x B) (A x C)

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11. Disjoint Unions

11.1 Definition:a. Two sets A and B are disjoint iff they

have no elements in common. In other words, A and B are disjoint A B =

b. A1,A2,…,An are mutually disjoint iff

i,j, Ai Aj =

c. {A1,A2,…,An } is a partition of A iff

i. A = A1 A2 … An

ii. A1,A2,…,An are mutually disjoint

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11. Disjoint Unions

Partitioning a set

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11. Disjoint Unions

11.2 Example: Let Z be the set of all integers.– Let A = {n Z | n = 3k for some integer k}– Let B = {n Z | n = 3k+1 for some integer k}– Let C = {n Z | n = 3k+2 for some integer k}

A = {…,-6,-3,0,3,6,…} B = {…,-5,-2,1,4,7,…} C = {…,-4,-1,2,5,8,…} A B = A C = B C = Z = A B C Therefore {A, B, C} form a partition of Z.

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12. Summary: Axiomatic Definitions Subset: A B iff x, xA xB Set Equality: A B iff A B B A Strict Subset: A B iff A B A B Union: x, x (A B) iff xA xB Intersection: x, x (A B) iff xA xB Difference: x, x (A B) iff xA xB Complement: x, x Ac iff xA Empty Set: (x, x {}) …or…(x, x A) A = {} Universal Set: (x, x U) …or …(x, x A) A = U Power Set: S, (S A) (S P(A)) Tuple Equality: (x1,…,xn) = (y1,…,ym) iff

n=m x1 = y1 x2=y2 … xn=yn

Cartesian Prod:(x,y) A B iff xA yB Disjoint Union: …

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Power sets, disjoint unions, ordered pairs and Cartesian Products are used in the lectures on Relations.

End of lecture