Matematika terapan week 2. set

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 2
SET THEORY

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.

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

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

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

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 }

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.

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”.
∈
∈

• {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”.

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

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.

Union
A∪B = { x | x ∈ A ∨ x ∈ B }
Elements in at least one of the two sets:
A B
U
A∪B

Intersection
A∩B = { x | x ∈ A ∧ x ∈ B }
Elements in exactly one of the two sets:
A B
U
A∩B

Disjoint Sets
DEF: If A and B have no common elements, they
are said to be disjoint, i.e. A ∩B = ∅ .
A B
U

Set Difference
A−B = { x | x ∈ A ∧ x ∉ B }
Elements in first set but not second:
A
B
U
A−B

Complement
A = { x | x ∉ A }
Elements not in the set (unary operator):
A
U
A

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 ∩=∪

Visual DeMorgan
A: B:

Visual DeMorgan
A: B:
A∪B :

Visual DeMorgan
A: B:
A∪B :
:BA∪

Visual DeMorgan
A: B:
A: B:

Visual DeMorgan
A: B:
A: B:
:BA∩

Visual DeMorgan
=
=∩ BA
=∪ BA

Matematika terapan week 2. set

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