matematika terapan week 4. fungsi dan relasi

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

TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Week 3

Relation and Function I


Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and Function

OverviewObviously, we do not realize that there many connections are happened in our circumtances. For examples, day and night happens because of earth rotation, all students in math are also connected to other subjects and so on. Strictly speaking, something happens because of other subject called “reason”.

Relations can be used to solve problems such as determining which pairs of cities are linked by airline flights in a network, finding a viable order for the different phases of a complicated project, or producing a useful way to store information in computer databases.

For couple weeks later, you all will be introduced this “connection” in mathematic’s view. And we shall learn to “map” or “transform” the “connection”.



Objectives

� Cartesian Product

� Relation

� Invers Relation

� Pictoral Repesentation of Relation

� Composition of Relation

� Relation Properties



Cartesian Product

Consider two sets A and B. The set of all ordered

pairs (a, b) where a∈A and b∈B is called the product, or Cartesian product, of A and B.

The short designation of this product is A x B,

which is read “A cross B”.


Relation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionRelation and FunctionEx.Let A={1, 2} and B={a, b, c}.

Then AxB {(1,a},(1,b),(1,c),(2,a),(2,b),(2,c)} BxA {(a, 1), (a,2), (b, 1), (b,2), (c,1),(c,2)} AxA {(1, 1), (1,2), (2,1), (2,2)}

From the example above we can conclude, that, First,

A x B ≠≠≠≠ B x AThe Cartesian product deals with ordered pairs, so naturally the order in which the sets are considered is important.

Second, using n(s) for the number of elements in a set S, we have

n(A x B) = n(A) . n(B) = 2 x 3 = 6Therefore, there will be 26 = 64 relation from A to B

So…..what is relation?????



RelationRelation is just a subset of the Cartesian product of the sets.

Definition.

Let A and B be sets. A binary relation or, simply, relation from A to B is a subset of A x B.

In other words, a binary relation from A to B is a set R of ordered pairs where the first element (domain) of each ordered pair comes from A and the second element (codomain or range) comes from B.



We use the notation a R b to denote that (a, b) ∈ R and a R b to denote that (a, b) ∉ R. Moreover, when (a, b) belongs to R, a is said to be related to b by R.

Assume C= {1,2,3} and D ={x,y,z} and let R {(1,y), (1,z), (3,y)}. Put the R or R for the followings:

1…X 1…Y 1…Z2…X 2…Y 2…Z

3…X 3…Y 3…Z

/

/



Invers Relation The invers relation of set is defined as the opposite mapping of relation itself.

Let R be any relation from a set A to a set B. The inverse of R, denoted by R-1, is the relation from B to A which consists of those ordered pairs which, when reversed, belong to R; that is,

R-1= {(b,a): (a,b) ∈ R}



Ex.

Let R = {(1,y), (1,z), (3,y)} from A = {1,2,3} to

B = {x,y,z}, then

R-1 = {(y, 1), (z, 1), (y,3)}



Pictoral Repesentation of Relation

Arrow Diagram



Table Representation



Matrice Representation

Suppose R is the relation from A to B, where

A={ a1,a2,a3,…,am} and B={ b1,b2,b3,…,bn}.

The relation can be describe in matrice M=[mij] as

folow:



Ex.

a1 = 2

a2 = 3

a3 = 4

b1 = 2

b2 = 4

b3 = 8

b4 = 9

b5 = 15



Directed Graph

First we write down the elements of the set, and

then we drawn an arrow from each element x to

each element y whenever x is related to y.

The point is, directed graph does not show the

relation between one set to the other. It just shows

the relation among the element inside the set.

Ex. R is relation on the set A = {1,2,3,4}

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



Prac.

Show the relation from

the directed graph

Bandung

Jakarta Surabaya

Medan

Makassar

Kupang



Composition of Relation

Suppose A, B, and C be sets, and let R be a relation from A to B and let S be a relation

from B to C. R ⊆ A x B and S ⊆ B x C.

Then R and S give rise to a relation from A to C, which is denoted by RoS and defined as



Ex.

Assume A= {1,2,3,4}, B ={a,b,c,d}, C ={x,y,z} and let R= {(1,a), (2,d), (3,a) (3,b), (3,d)} and S ={(b,x), (b,z), (c,y), (d,z)} . Show the relation a(RoS)c!



From the picture we can observe that there is an arrow from 2 to d which is followed by an arrow from d to z. We can view these two arrows as a “path” which “connects” the element 2 ∈ A to the element z ∈ C. Thus,

2(R o S)z since 2Rd and dSz

Similarly there is a path from 3 to x and a path from 3 to z. Hence,

3(R o S)x and 3(R o S)z

No other element of A is connected to an element of C. Therefore, the composition of relations R o S gives

RoS= {(2,z), (3,x), (3,z)}



Soal :

R = {(1, 2), (1, 6), (2, 4), (3, 4), (3, 6), (3, 8)}

S = {(2, u), (4, s), (4, t), (6, t), (8, u)}

Gambarkan grafiknya dan tentukan R o S



R o S = {(1, u), (1, t), (2, s), (2, t), (3, s), (3, t), (3, u) }



Exercises :

1



2.

matematika terapan week 4. fungsi dan relasi

Education

binary relation

na x b

eng matematika terapan

ilyas hadikusuma

set r of ordered pairs

pairs of cities

applied math

element domain