2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Week 2

SET THEORY

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

OVERVIEW

Why do we learn about set?

Localizing a system into groups would make the system itself simple to understand and to redesign.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

OBJECTIVES:

1. Set definition and its properties

2. The representation methods of set

3. Set operation

4. Subset, Power of Set, Venn diagram

5. Computer representation of Set

6. Cartesian Product

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

Definition

Set is a group of an unordered different or unique object.

The object of set named as element or member.

WE USE CAPITAL LETTERS TO REPRESENT THE SET

we use lower letters to represent the elements

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORY

Examples

A = Set of three PC’s hardware which has area more than

5 x 5 cm2. B = Set of 5 non living things in this math class

C = Set of person using spectacle in Math class

We can define the elements using bracket { }. Thus,

A = { keyboard, monitor, motherboard }B = { tables, chairs, infocus, spidols, laptop }

C = please help me to define them …… ;P

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

The representation methods of set

There are 2 methods to represent the set.

1. List all of the members

2. State the member properties

Ex.

1. A = { keyboard, monitor, motherboard}

2. B = { x | x is a positive odd interger, x < 10 }

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

Exercises :

Express the following set into form of set member properties (2nd method)!

1. The set B is natural number more than 3 and less or equal to 15.

2. The set of C is real number more than or equal to -5 and less than 10

3. The set of D is even number of interger less than 20.

Express all of above in 1st method as well.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

Set notation

• The simplest example of a set is Ø, the empty

set, which has no members or elements

belonging to it.

• x Y means “x is a member of Y”.

• x / Y means “x is not a member of Y”.

• {x, y} means “the set whose members are x and

y”.

∈

∈

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

• {x | P} means “the set of all values of x that have the property P”. For example,

{ x | x is an even integer } is the set of all even integers.

• X = Y means “X and Y have the same

members”.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

Set Operation / Manipulation

Set theoretic operations allow us to build new sets

out of old, just as the logical connectives allowed

us to create compound propositions from simpler

propositions.Several types operation of set are

1. union

2. intersection

3. disjoint set

4. difference of set

5. complement

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

SET THEORYSET THEORYSET THEORYSET THEORY

Venn diagrams are useful in representing sets and set operations. Various sets are represented by circles inside a big rectangle representing the universe of reference.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Union

A∪B = { x | x ∈ A ∨ x ∈ B }

Elements in at least one of the two sets:

A B

U

A∪B

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Intersection

A∩B = { x | x ∈ A ∧ x ∈ B }

Elements in exactly one of the two sets:

A B

U

A∩B

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Disjoint Sets

DEF: If A and B have no common elements, they

are said to be disjoint, i.e. A ∩B = ∅ .

A B

U

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Set Difference

A−B = { x | x ∈ A ∧ x ∉ B }

Elements in first set but not second:

A

B

U

A−B

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Complement

A = { x | x ∉ A }

Elements not in the set (unary operator):

A

U

A

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Set Identities via Venn

It’s often simpler to understand an identity by drawing a Venn Diagram.

For example DeMorgan’s first law

can be visualized as follows.

BABA ∩=∪

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

A: B:

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

A: B:

A∪B :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

A: B:

A∪B :

:BA∪

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

A: B:

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

A: B:

A: B:

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

A: B:

A: B:

:BA∩

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Visual DeMorgan

=

=∩ BA

=∪ BA