Relation

Relation

Indonesia consist of land and water. Indonesia have so many big island such as Sumatra, Java, and Kalimantan. On each island consist of so many town such as Medan, Palembang, Jakarta, Surabaya and Pontianak. Medan in Sumatera island. Now, discuss with your friend with your friends on group about another country. How to show the location of the town?

Set A={Sumatra, Java, Kalimantan}Set B={Medan, Palembang, Jakarta, Surabaya, Pontianak}

Medan and Palembang located on Sumatra

Jakarta and Surabaya located on Java

Pontianak located on Kalimantan

Located on is relation which joint each city and the island

Definition of relationA relation from Set A to Set B

is a regulation which connects the members of Set A to the members of Set B.

How to show relation?

By using arrow diagramBy using ordered pairsBy using Cartesian coordinate

Showing relation by using arrow diagram• Create two ovals with the elements

of first set on the left and the elements of second set on the right.

• Elements are not repeated. (when you find one element raise more than 1 time. Write only once)

• Connect elements of first set with the corresponding elements in the second set by drawing an arrow.

Example

Medan Palembang Sumatra

located on

JakartaSurabaya

Java

Pontianak Borneo

Showing Relation by Using Ordered Pairs

Order = urutanPairs = pasanganOrdered pairs arrange the pairs

well.Put each element which relate to

another element on bracket as a pairs, and all pairs put in one parenthesis

Example {(Medan, Sumatra),(Palembang,

Sumatra),(Jakarta, Java),(Surabaya, Java),(Pontianak, Kalimantan)}

Showing Relation by Using Cartesian Coordinate

All elements of first set put on x-axisAll elements of first set put on y-axis

If given the number of two different sets, how many it possible relation?

Number of possible relation from two different sets called Cartesian product

Suppose:Number of elements of set A is aNumber of elements of set B is bNumber of Cartesian product can

be determine by axb

Example A={1,2,3,4,5}B={1,2,3}Number of possible relation from set

A into set B?n(A)=number of element of set An(A)=5n(B)=3Number of possible relation from set

A into set B = n(A)xn(B)=5x3=15

Exercise

a.Identify the elements of set A and set Bb.Name the relationc.Show the relation by using ordered pairsd.Show the relation by using Cartesian coordinate

1.

2.

a.Identify the elements of set P and set Qb.Name the relationc.Show the relation by using ordered pairsd.Show the relation by using arrow diagram

Name the RelationChoose one name for that

relation, and check for all connection you find.

If all connection fulfill that name, so name you choose is the correct relation

Important WordDomain = daerah asalCo-domain = daerah kawanRange = daerah hasilFunction = fungsi = pemetaan

Indicator of DomainArrow diagram : all elements on the

first oval(set)Ordered pairs : all first elements on

each pairsCartesian coordinate : all elements

on x-axis

If the elements of each set raised more than 1 time, you only write that element once.

Indicator Co-domainArrow diagram : all elements on

the second set (oval)Ordered pairs : all second

element on each pairsCartesian coordinate : all

elements on y-axis

Indicator RangeArrow diagram : all elements of

co-domain which have relation to domain

Ordered pairs : all second element of each pairs

Cartesian coordinate : all element of co-domain which have relation to domain

Example

Domain = {salt, sugar, vinegar, pepper}Co-domain = {sour, salty, bitter, sweet, hot}Range = { sour, salty, sweet, hot}

Type of RelationOne to one relationOne to many relationMany to one relationMany to many relation

How to identify which Graph is function

Graphs of a FunctionVertical Line Test:

If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.

x

y

x

y

Does the graph represent a function? Name the domain and range.

YesD: all realsR: all reals

YesD: all realsR: y ≥ -6

x

y

x

y

Does the graph represent a function? Name the domain and range.NoD: x ≥ 1/2R: all reals

NoD: all realsR: all reals

Does the graph represent a function? Name the domain and range.YesD: all realsR: y ≥ -6

NoD: x = 2R: all realsx

y

x

y

The number of Function of two Sets

If the number of element sets A is n(A) = a and the number of element sets B is n(B) = b, so:

a) The number of the possible function of sets A to B = (b)a

b) The number of the possible function of sets B to A = (a)b

ExampleGiven A = {4,5,6} and B = {3,5}Determine the number of the possible function

of:a) A to Bb) B to A

AnswerA = {4,5,6} n(A) = a = 3B = {3,5} n(B) = b = 2So:a) The number of function of A to B = (b)a = 23 = 8b) The number of function of B to A = (a)b = 32 = 9

Correspondence One to One1) Definition : Correspondence one to one of

sets A and sets B is the relationship which relates every member of set A to exactly one member of set B and relates every member of set B to exactly one member of set A.

The number of elements sets A and sets B are equal

2) The number of correspondence one to one

If n(A)=n(B)=n,the number of possible correspondence one to one A and B is;

n x (n-1) x (n-2) x….3x2x1

exampleHow many the number of

corraspondence one to one between sets P and Q, if P = {a, b, c, d} and Q = {3, 5, 7, 9}

answern(P) = 4 and n(Q) = 4The number of correspondence

one to one P and Q= 4 x 3 x 2 x 1= 24 ways

Number of one to one correspondence is 720.What is the numbers of each elements of each set which construct that correspondence?

n=6

Function Notation• When we know that a relation is a function, the “y” in the equation can be replaced with f(x).

• f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’.

• The ‘f’ names the function, the ‘x’ tells the variable that is being used.

Symbol the functionIf we have function g and map x

into x2-2, we symbol:g : x x2-2 or g(x) = x2-2 ory = x2-2

Value of a FunctionSince the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2.

Find f(4):f(4) = 4 - 2f(4) = 2

Value of a FunctionIf g(s) = 2s + 3, find g(-2).

g(-2) = 2(-2) + 3 =-4 + 3

= -1 g(-2) = -1

Value of a FunctionIf h(x) = x2 - x + 7, find h(2c).h(2c) = (2c)2 – (2c) + 7

= 4c2 - 2c + 7

Value of a Function

If f(k) = k2 - 3, find f(a - 1) f(a - 1)=(a - 1)2 - 3 (Remember FOIL?!) =(a-1)(a-1) - 3 = a2 - a - a + 1 - 3 = a2 - 2a - 2