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Additional mathematics

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Prepared by : PN HJH SARIPAH AHMAD Sekolah Menengah Sains Muzaffar Syah Melaka 75450 AYER KEROH MELAKA

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LEARNING AREA : FUNCTIONS Learning Objectives : Understand the concept of relations Learning Outcomes : Student will be able to

1.1 Represent relation using a) arrow diagram , b) ordered pairs, c)graphs 1.2 Identify domain, co domain, object, image and range of a relation. 1.3 Classify a relation shown on the mapped diagram as : one to one many to one , one to many or many to many

1.1a Representing a relation between two sets by using an arrow diagram Example 1 Suppose we have set A = { 0,1,2,3 } and set B = { 0,1,2,3,4,5,6} .Let us examine the relations “is one less than “ from set A to Set B . One way to show the relations is to draw an arrow diagram as shown below . The arrows relate the elements in A to the elements in B

A

is one less than

B

•6 •5

In the space below give other example to show a relation

3 2 1 0 .

FIG 1.1

4 3 2 . 1 •0

A relation from set A to set B is an association of elements of A with elements of B . Exercise 1 1 Two set of numbers, P and Q are shown below Complete the arrow diagram to show the relations “ is grater than “ from set P to set Q P is grater then •

2. What relation from set S to set T is illustrated in diagram below .Write your answer in the box below

Q

10 8 6 1 . .

8 3 2 1

3 2 1 -1 .

6 4 2

•1 -2

3. Construct an arrow diagram to show the relation “ is a factor of “ from set A ={2,5,7,13 } to set B = {1,4,15,35,40}

4. Construct an arrow diagram to show the relation “ is a multiple of “ from set A ={1,2,3,4,5,6 } to set B = {2,3,5}

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1.1(b) Representing a relation between two sets by using an ordered pairs The relation can also be shown concisely by ordered pairs ( 1, 2 ) . The elements in the pair are in order since the first element comes from the first set A and the second element comes from the second set B .We have A = {0,1,2,3} and B = {0,1,2,3,4,5,6, If x ∈ A ( this means that x is a member of A) and y ∈ B, the set of all ordered pairs (x,y) is { (0,1),(1,2),( 2,3),(3,4)} . This set of ordered pairs defines the relations “ is one less than “ from set A to set B Example 2 1. A = { 1,2,3,4,5,6} and B = {2,3,5 }. Show the relation “ is a multiple of” from A to B as a set of ordered pairs Solution The ordered pairs ={ (2,2),(3,3),(4,2),(5,5),(6,2),(6,3) } 2. Write down the relation as a set of ordered pairs “ is factor of “from set A ={2,5,7,13 } to set B = {1,4,15,35,40} Solution The ordered pairs = { (2,4) (2,40), (5,15),(5,35),(5,40),(7,35) }

Exercise 1.1 (b)

1. A relation between two sets is defined by the set of ordered pairs , { (-1,2),(1,4) ,(3,6) (5,8) , (7,10 ) } List the elements of the two sets, and describe in words a possible relation between the first set and the second set 2. Given that A = {2,4,6,8} and B = { 1,2,3,4,5 } If x ∈ A and y ∈ B , list the set of ordered pairs in the relation x “ is double “ y

3. Given that X = { 0,1,2,3 } and Y = { 0,1,2,3,4,5,6} If x ∈ X and y ∈ Y , list the set of ordered pairs in the relation x “is one less than “ y

4. A relation R is defined by { (

1 1 ,8),(1,4),(2,2),(4,1) (8, )} 2 2

List the set of first members of pairs, and set of second members of the pairs . Describe in words a possible relation between the first set and the second set.

1.1(c)

Representing a relation between two sets using a graphs

We can plot an ordered pairs in a Cartesian graph of the relations Example 3 1. We have A = {0,1,2,3} and B = {0,1,2,3,4,5,6, If x ∈ A and y ∈ B, the set of all ordered pairs ( x,y) is (0,1),(1,2),(2,3),(3,4)} we can plot a Cartesian graph to show the relation. 2....

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