set presentation note

8/3/2019 Set Presentation Note

1/114

FOUNDATIONMATHEMATICSGROUPTEN

SUMMARIZEOFOVERALLSYLLABUS

MUHAMAD ARIF BIN NASARUDDIN D20091035123

HASNOR IZZATI BT CHE RAZALI

D20091035102MOHD AIDIL UBAIDILLAH BIN RAZILAN

D20091035132

NORHAMIDAH BT ROHANI

D20091036643

NUR HAFIZAH BT ZAHARI

D20091035133

NOR SHARLIDA BT MOHAMAD JALAILUDIN

D20091035098


2/114

CHAPTER 1

SET


3/114

Set is collection of elements.

Venn Diagram can represent set :


4/114

DO YOU STILLREMEMBER?

Q : n(A) represent of what? A : Number of element of set

Q : and { } indicate of what? A : Empty set

Q :How we can say two set are equal set? A : If element in both set are equal

Q : What is subset? A : When all element in one set contain in other set


5/114

TYPEOFSUBSET

Proper subset

Symbol :

Example : C B A

Improper subset

Symbol :

Example : D E

D D=EE


6/114

From the diagram identify :

The Universal SetClue > set that contains all elements in the discussionAnswer = { 4,3,8,9,22,10,13 }

The element of complement Set for set AClue > Set that contain all elements in universal set

which are not element in AA = { 9,10,22,13}

Set of intersection between two set(A B)Clue > all elements are common elements for both set A B = { 4,3 }

Element of union of two set(AB)Clue > every elements in both setsAB = { 3,4,8,9,10 }


7/114

ADDITIONAL PRINCIPLE

Two sets

n (A B) = n(A) + n(B)n (A B)

Three sets

n (ABC) = n(A) + n(B) + n(C) -n (A B) -

n (A C) -n (B C) + n (A B C )


8/114

CARTESIANPRODUCT

Donated by A x B

Example

If A = {1,2} and B = {d,c}. What is element of Ax B ?

Answer

(A x B) = {(1,d),(1,c),(2,d),(2,c)}


9/114

CHAPTER 2

REAL NUMBER SYSTEM


10/114

CHAPTER 2 : REALNUMBER SYSTEM

Real Number Set

(R) Q Q

Ratio Set Number(Q)

An Integer numberthat can be dividedby 1. Eg : 5,7,9Decimal endednumber. Eg : 1.25

Decimal numberrepeated. Eg :0.11111

Non Ratio Set

Number (H or Q)Decimal numbersare and notrepeated.Eg :0.56783315678

Real Set Number(W)

W = {0,1,2,3,}

Integer (Z)Z = {-2,-1,0,1,2,}

Natural Numbers(N)

Count numberN = {1,2,3,4,}

Prime Number Set

positive integernumber except 1that can be dividedby 1 and its ownnumber

Composite NumberSet

Positive numberexcept 1 and not aprime number


11/114

NUMBERLINE

Represent all real number.

Example :


12/114

ABSOLUTEVALUE

a Equal to

a when a 0

-a when a < o


13/114

PROPERTYOFREALNUMBER

Type of Property Explanation

Closure A + B = C(A)(B) = CWhere C is real number

Commutative C + D = D + C(C)(D) = (D)(C)

Associative (A + B) + C = A + (B + C)(AB)(C) = (A)(BC)

Distributive A(B + C) = AB + ACA(B C) = ABAC

Identity C + 0 = 0 + C = CD(1) = (1)D = D

Inverse A + (-A) = 0 = (-A) + A


14/114

INEQUALITY

Symbol Meaning

> More than

< Less than

More than or equal

Less than or equal


15/114

INTERVAL

Closure interval

{ x I a x b } = [a,b]

Open interval

{ x I a < x < b } = (a,b)

Half open interval and half closure interval

{ x I a x < b } = [a,b)

Infinite interval

{ x I x > a } = (a,+ )


16/114

EXPONENT,LOGARITHM ANDRADICAL

CHAPTER 3:


17/114

DEFINITION 1

(positive integer) , For all a R and n Z

........

n factors of a

a a a a naa to the power of n

a = base

n = exponent or power or index

an = an exponential expression


18/114

DEFINITION 2

,For all a R and n Z

1 , 0

... , 0

1

n

n

n

n

a n

a a a a a n

aa


19/114

DEFINITION 3

If a is a real number, mand nare integers,

n

m

a

mn

a

n m

a

Ratio index


20/114

1.

2.nm

n

m

a

a

a

m n m n

a a a


21/114

3.

mnnm aa

nnn

baab )(

n

nn

b

a

b

a

4.

5. , 0b


22/114

DEFINITION

If a, n R+ and n = a x, then log a n = xwhere a 0.

If a=10, log 10 n = x

log 1b

b

log xb n x n b

LOGARITHM


23/114

Laws of logarithm

If a, M, N

R+ and p

R, then

a)

MNalog log loga aM N

p

a Mlog Mp alogc)

b)

N

Malog log loga aM N


24/114

DEFINITION

Radical or surd is a non-ratio number in theform where n, b are real and b > 0.

a) MULTIPLICATION OF RADICALb) DIVIDING OF RADICAL

c) ADDITION AND SUBTRACTION OF RADICAL

d) DENOMINATORS THAT HAS RADICAL

n b


25/114

CAUTION

x bab a

a ab b

c ( ) a b b a c b


26/114

TORATIONALIZEEXPRESSIONSINVOLVINGSURDS When surds occur in quotients, it is

customary to rewrite the quotient so that

the denominator is not in surds form. This

process is called rationalization of

denominator.


27/114

RATIONALIZING THEDENOMINATOR

Rationalize the denominator means thedenominator contains no square roots.

Rationalizingthe multiplierwith theconjugate of the original denominator.

In rationalizing the denominator of a

quotient, be sure to multiply both thenumerator and the denominator by thesame expression.


28/114

CONJUGATE

a b a b

a b a b


29/114

CHAPTER 4

COMPLEX NUMBER

T


30/114

THEIMAGINARYUNITI

The imaginary unit iis defined as i= , where i2= -1

1

Using the imaginary unit i, we can expressthe square root of any negative number asa real multiple of i.

For example,

16 16i = 4i


31/114

Any number of the form z= a + bi , thereforea is called

the real part of the number and bis called the imaginarypart of the number .

The sum of a real number and imaginary numberproduces a COMPLEX NUMBER

Definition


32/114

EQUALITYOFTWOCOMPLEXNUMBERS

For two complex numbers = a+ biand= c+ di. Therefore, if

a= cand b= d.

1z

2z

21 zz


33/114

COMPLEX CONJUGATEOFACOMPLEX

NUMBER

The complex conjugate of a complex number, a + bi isa bi and a conjugate of a bi is a + bi.

The multiplication of complex conjugates gives a real

number.

(a + bi) (a - bi) = a2 + b2

(abi) (a+ bi) = a2 + b2

This fact is used to simplify the expressions where thedenominator of a quotient is complex.

ADDING AND SUBTRACTING COMPLEX


34/114

ADDINGANDSUBTRACTINGCOMPLEX

NUMBERS

Complex numbers can add together by adding the real

parts and then adding the imaginary parts. You can subtract one complex number from another by

subtracting the real parts and then subtracting theimaginary parts.

So:(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) (c + di) = (a c) + (b d)i


35/114

MULTIPLYINGONECOMPLEXNUMBERBY

ANOTHER

To multiply two complex numberstogether, apply the rules of algebra.

So :(a + bi) (c + di) = ac + adi + bci+ bdi

= ac + (ad+ bc)ibd

= (acbd) + (ad+ bc)i


36/114

ARGAND DIAGRAM

Any complex number z = a + bican be represented by any

ordered pair (a, b) and hence plotted on xy-axes with the

real part measured along x-axis and the imaginary part

along they-axis.The graphical representation of the

complex number field is called an Argand diagram.

T OA


37/114

A (a,b)

THELENGTH OA ISCALLEDTHEMODULUSOFTHECOMPLEXNUMBERA + BIANDISWRITTENA + BISOTHAT

A + BI= Z =

z

Im (y)

Re (x)

O a

b

22 )()( ba

Th l i ll d h f bi d i
http://c/KMPerlis/My%20Documents/NUMBER%20SYSTEM/LECTURE%20%207%20OF%207.doc


38/114

The angle is called the argument of a + biand iswritten

1tan

b

a

Arg (a + bi) = = ,

QUARTER 2 QUARTER 1

= + =

QUARTER 3 QUARTER 4

= - =

1tan

b

a

1tan b

a

1tan

b

a

1

tan

b

a

180o

180o


39/114

POLAR FORM OF A COMPLEX NUMBER

Given that z= a+ bi, as shown:

a= |z|cos

b= |z| sin

|z|

x

(a,b)

y

| | (| | i )i


40/114

z = |z| cos + (|z| sin )i

= |z| (cos + isin )

For clarity, we write |z| as r. Thus, z = a + bican be written as;

z= r(cos + isin )

This is called the Polar Form of a + bi. In contrast, a + biiscalled the Cartesian Form of z.


41/114

COORDINATE

GEOMETRY

CHAPTER 5


42/114

CARTESIAN COORDINATESYSTEM


43/114

DISTANCEBETWEENTWOPOINT

By using Pythhogoras Theorem:

PQ2 = PR2 + RQ2

(x2,y2)

(x1,y1)

(x2,y1)R

d

Q

P

d =


44/114

MIDDLE POINT COORDINATE

Midpoint (m)

m=


45/114

DIVIDINGPOINTWITH RATIO

Inside Dividing point Outside Dividing point

m

nB(x2 , y2 )

A(x1 , y1 )

P=(x , y )m n

A(x1 , y1 )

B(x2 , y2 )

Q=(x , y )


46/114

GRADIENTLINEQ(x

2,y

2)

P(x1, y1 )

m

m=

Positive gradient Negative gradient

Wherex2 x1


47/114

STRAIGHT- LINEEQUATION

Gradient Type

Interception Type

General Type

y = mx +c

ax+by+c =0

m= gradientc= interception

at y-axis

a= interception

at x-axisb= interception

at y-axis

With a,b,cconstant


48/114

THENEARESTPOINTTOSTRAIGHTLINE

ax+bx+c=0

(h,k)

Q

P


49/114

TRIANGLESANDSQUARESAREA

Triangles Area Squares Area


50/114

CHAPTER 6

FUNCTIONANDGRAPH

52

RELATION AND FUNCTION


51/114

53

RELATION AND FUNCTION

Types of Relation

There are 4 types of relation :

i) One to one- each element in set X is connected to an element in set

Y

1

2

3

1

4

9

X Y

>

>

>

is the square of

(ii) M


52/114

54

(ii) Many to one

0

1

4

-2

-1

1

2

Y

X is the square root of

>>

>

>


53/114

55

(iii) One to many

(iv) Many to many

a

b

c

d

e

f a

bc

d

1

2

3

4

F i


54/114

56

Function

A function is a special case of a relation which

takes every element of one set (domain) andassigns to it one and only one element of asecond set (range).

Therefore,

i) one to one

ii) many to oneRelations are function


55/114

57

Vertical Line Test :

To test if a graph displayed is a function.

The graph is a function if each vertical line drawn through the domaincuts the graph at only one point.

Vertical lines are drawn parallel to the y-axis

Domain and Range


56/114

58

Domain and Range

Domain of f(x) is the set of values of x for which f(x) is defined.

Range of f(x) is the set of values of yfor which elements in the domainmapped.

We can evaluate the domain and range by :(i) Graph (ii) Algebraic approach

Basic shape of a function


57/114

59

Basic shape of a function

(i)Quadratic function

x0

a) f(x) = x2 b) f(x) = -x2

f(x)

x

0

f(x)


58/114

60

(ii) cubic function.

a) f(x) = x3

f(x)

b) f(x) = -x3

f(x)

xx0 0

(iv) Reciprocal function


59/114

61

f(x)

x0

a) y =

f(x)

x0

b) y =1

x

1

x

(v)Absolute value function |f(x)|


60/114

62

(v)Absolute value function, |f(x)|

0x

f(x)

0x

f(x)

2

1 2

b) f(x) = |x2-3x + 2|a) f(x) = |x|


61/114

63

ADD, SUBTRACT, DIVIDEAND MULTIPLYTWO

FUNCTIONS.

We can combine two or more functions in anumber of ways.

Addition

f(x) + g(x) = g(x) + f(x)

Substraction

f(x) - g(x) g(x) - f(x) Multiplication

f(x)g(x) = g(x)f(x)


62/114

64

This can be represented in anarrow

diagram:

x g(x) f[g(x)]

g f

gf


63/114

65

This can be represented in anarrow

diagram:

x f(x) g[f(x)]

f g

INVERSE FUNCTION


64/114

66

INVERSE FUNCTION

If f:x y is a function that maps x to y, then the inverse function is denoted

by where

is a function that maps y back to x.

x y

Note :

1

( )f x

1f 1f

1f

f

1f


65/114

67

An inverse function of f exists only if the function fis a

one to one function.

Method to test whether a function is 1-1 :

i. Algebraic approach.If ,

then1 2( ) ( )f x f x

1 2x x

ii) Horizontal line test ( graphical approach)


66/114

68

If the horizontal line intersects the graph of the function only once , then

the function is one to- one.

one-to-one Not one-to-one

ii) Horizontal line test ( graphical approach)

a) b)

The horizontal lineintersects

the graph at one point.

The horizontal line

intersectsthe graph at two points.

Method Of Finding The Inverse Function


67/114

69

and

Method Of Finding The Inverse Function

1) Using Formula

2) By substitution

1

[ ( )]f f x x

1 1

1

1 1 1

( )

[ ( )]

( )

f f

f f x x

f g g f

1[ ( )]f f x x


68/114

CIRCLE

Standard Equation of center (0,0).

Standard Equation of center (h,k).

General Equation.

Radius =

Center =

2 2 2x y r

2 2 2

( ) ( )x h y h r

2 2 2 2 0x y gx fy c

( , )g f

2 2r g f c


69/114

TRIGONOMETRY

CHAPTER 7

TRIGONOMETRIC RATIOS AND IDENTITIES


70/114

Generally the diagram above can be formulated as

the diagram below

P

R

QR

PQ

PR

QR

PR

PQ

adjacent

oppositetan

hypotenuse

adjacentcos

hypotenuse

oppositesin

Q

TRIGONOMETRIC RATIOS AND IDENTITIES


71/114

From the diagram:

cosec = =

sin

1

y

z

sec =cos

1 =xz

cot =tan

1=

y

x


72/114

sin (900 - ) = cos

900 - z

y

x

cos (900 - ) = sin

tan (900 - ) = cot

Trigonometric Ratios of Particular Angles


73/114

EQUILATERALTRIANGLEOFSIDES 2 UNITINLENGTH

sin 600 =2

3

cos 600 =21

tan 600 =

1

3

sin 300 =2

1

cos 300 =23

tan 300 =3

1

3

600

2 2

1 1

600

300

Trigonometric Ratios of Particular Angles


74/114

ISOSCELESTRIANGLE

sin 450 =2

1

cos 450 =2

1

tan 450 = 1

450

4502

1

1

POSITIVE ANGLE


75/114

POSITIVE ANGLE

sin All

cos tan

TRIGONOMETRIC IDENTITIES

cos2 + sin2 = 1

1 + tan2= sec2

cot2 + 1 = cosec 2


76/114

COMPOUND ANGLE

BsinAcosBcosAsinBAsin

BsinAsinBcosAcosBAcos

BsinAcos-BcosAsinBAsin

cos A B cos A cos B sin A sin B

BA tantan1BtanAtan

BAtan


77/114

DOUBLE ANGLE

AcosAsin22Asin

A2sin-1

1-A2cos

Asin-Acos2Acos

2

2

22

Atan-1

Atan22Atan

2


78/114

FACTOR FORMULAE

2

B-Asin

2

BAsin2BsinAsini.

2

B-Acos

2

BAsin2BsinAsinii.

A B A - Biii. cos A cos B 2 cos cos

2 2

2

B-Asin

2

BAsin2BcosAcosiv.


79/114

CHAPTER 8

POLYNOMIALSANDRATIONALFUNCTION


80/114

8.0 POLYNOMIALS

8.1 Polynomials 8.3 Partial

Fractions

8.2 Remainder Theorem,

Factor Theorem and Zeroes of

Polynomials

POLYNOMIALS


81/114

POLYNOMIALS1

1 1 0( ) ... ; 0n n

n n nP x a x a x a x a a

n ZWhere the coefficients

are real numbers and

1 2, , ,...,

0 na a a a


82/114

MONOMIAL,BINOMIAL,POLYNOMIAL.

Name Example

Monomial

Binomial

Polynomial

3

2x x

4x

25 2 1x x


83/114

OPERATIONOFPOLYNOMIAL

4 3 2( ) 2 3 1f x x x x

4 3 2

( ) 1f x x x x

4 3 2( ) 3 2 4 2f x x x x 4 2( ) 2f x x x

Addition and subtraction:

ADDITION SUBTRACTION

For example, given that,

Multiplication


84/114

86

IfP(x) is a polynomial of degree m and

Q(x) is a polynomial of degree n,

then product

P(x)Q(x) is a polynomial of degree (m + n)

Note

Multiplication( ) ( )P x Q x


85/114

Division

synthetic division:ex: P(x)= , ( )

2 2 5 8 17 6

0 4 2 20 6

2 1 10 3 0

Remainder

If zero , divider is the root.

use its root to divide the function

4 3 22 5 8 17 6x x x x 2x


86/114

long division,

ex:

4 3 22 2 5 8 17 6x x x x x

Use its factor to divide the

function.

3 2

2 9 10 3x x x

4 32 4x x3 29 8 17 6x x x

3 29 18x x

210 17 6x x 210 20x x

3 6x 3 6x


87/114

89divisor (quotient) + remainder

the division of

polynomials can be expressed in the form

orP(x R( x )

R( x )D

)

D(x)P( x )

Q( x )

Q( x )( x )

D( x )

8 2 REMAINDER THEOREM FACTOR THEOREM


88/114

8.2 REMAINDER THEOREM, FACTOR THEOREMAND ZEROES OF POLYNOMIALS

Generally,when is divided byx-a,let the quotient beQ(x) and the remainderR

then,

substitute ,

this the remainder theorem,

when a polynomial P(x) is divided byx-a, the remainder

is P(a).

( )P x

x a ( ) 0P a R ( )R P a


89/114

From the remainder theorem, the remainder when P(x) isdivided byx-a is P(a).

ifP(a)=0 then the remainder is zero.

so, P(a)=0 x-a is a factor ofP(x)

is the factor theorem where a is called a zero ofP(x)=0.


90/114

8.3 Partial fractions

a)linear factors

b)quadratic factors

c)repeated factors

2 3

(3 4)( 2) 3 4 2

x A B

x x x x

2 2

2 3

(3 4)( 2) 3 4 2

x A Bx C

x x x x

2 2

2 3

(3 4)( 2) 3 4 2 ( 2)

x A B C

x x x x x


91/114

RATIONALFUNCTION

The quotient of 2 polynomial functions:

Where, is the domain of function.

( )( )

( )

g xf x

h x

( )h x

HOW TO SKETCH THE RATIONAL FUNCTION


92/114

HOWTOSKETCHTHERATIONALFUNCTIONGRAPH?

From the , determine:

Domain of the graph

Vertical asymptotes

Horizontal asymptotes Get the value of from right and left domain,

which are:

or

Then, sketch the graph

( )( )( )

g xf xh x

( ) 0h x

( )f x

( )f x ( )f x

1. Equal Vectors


93/114

Two vectors are equal if they have the samemagnitude and the same direction

If a is any nonzero vector, then a , the negative of a isdefined to be the vector having the same magnitude as a butoppositely directed.

a

-a

2. Negative Vector

3. Addition Of Vectors


94/114

If a and b are any two vectors, then the sum a+b is the vectordetermined as follows :

Position the vector b so that its initial point coincides with theterminal point of a. The vector a+b is represented by the arrowfrom the initial point of a to the terminal point of b.

b

a

a+b

A

B

AB a b = resultant vector

CHAPTER 9


95/114

CHAPTER 9

VECTOR

Magnitude of a vector


96/114

:The magnitude of = ai + bjis

2 2v a b

v

6. Modulus of a vector


97/114

The modulus (length) of a vector is its magnitude

The modulas of vector ABis denoted by the

Symbol .The length (modulus) of vector is

denoted by the symbol .

AB aa

7. Zero Vector


98/114

The zero vector, donated by 0, has magnitude zero.

Contrary to all the vector, it has no specific direction

Two non-zero vector and parallel if one is a scalar

multiple of the other,That is , is a scalar.

8. Parallel vector

a b

a b

1. Vector In Two Dimension


99/114

The position vector of any point P(a, b,)is ai + bj

P

x

y

Vector Operation


100/114

Vector Operation

i) Magnitude : The magnitude or length of

is

ii) Additionand : If and ,then

Subtraction

=(a1 + a2)i + (b1 + b2)j=(a1 a2)i + (b1b2)j

iii) Scalar multiplication : If kis a scalar and

= ai + bj a vector,then

k = (ka)i + (kb)j

2 2v = a b

v a b i j

1 1 1v a b i j 2 2 2v a b i j

1 2v v

1 2v v

v

v

iv) Unit Vector


101/114

) U t ectoIf =xi + yjthen,the unit vector in thedirection of

Is2 2

a x ya

a x y

a

a

i j

2. Vector In Three Dimension


102/114

x

y

z

The position vector of any point P(a, b, c)is ai + bj + ck

a

b

c

P(a, b, c)

Vector operation


103/114

i) Magnitude : The magnitude or length of is =

ii) Additionand : If = a1i + b1j + c1k and = a2i + b2j + C2k ,thenSubtraction=(a1 + a2)i + (b1 + b2)j + ( c1 + c2)k

=(a1 a2)i + (b1b2)j + ( c1c2)k

iii) Scalar multiplication : If nis a scalar and = ai + bj +ck a vector,thenn = (na)i + (nb)j + (nc)k

iv) Unit Vector : if then,The unit vector in the direction

Is

2

1

2

1

2

1 cba

a x y z

2 2 2a x y zaa x y z

vv

1v 2v1 2v v

1 2v v

vv

i j k

i j k

SCALAR PRODUCT


104/114

The scalar product(dot product) of two vectors

and is denoted by and defined as

Where is the angle between and

a b

cosa b a b

a b

a b


105/114

Rule 1

If two vector and are parallel, thenthe angle between the two vectors iseithor 0o or 180o (i.e the vector are

parallel and in the same direction )

a b

Rule 2


106/114

For unlike parallel vector (i.e the vector are parallel but in

the opposite direction)

b

ba

a

Angle between and is

a

b

0

180a b0cos180a b a b

a b

Rule 3


107/114

If two vector a and b are perpendicular, then the angle between

the two vector is 90, therefore

cos90oa b a b

0

a

b


108/114

Properties of Scalar

Product

a b b a

( ) )a b c a b c a b c

2

a a a

( )a b c a b a c

( ) ( ) ( )m a b ma b a b m

a b a b if and only if parallel to ba

if and only if is perpendicular to0a b a b

VECTOR PRODUCT


109/114

The vector product is also known as cross product

because the result of the vector product is a vector.

Let and are two vector in a plane. Vector product

defined as

Where is the angle between and

a b

sina b a b

a b

VECTOR PRODUCT

Let ,1 1 1a x y z b x y z i j k


110/114

1 1 1

2 2 2

1 1 1 1 1 1

2 2 2 2 2 2

a b x y z

x y z

y z x z x yy z x z x y

1 2 2 1 1 2 2 1 1 2 2 1y z y z x z x z x y x y

1 1 1a x y z 2 2 2b x y z i j k

i j k

i

i j k

j k


111/114

Let , and are non-zero vectors and is a

scalar.

a b c

1.

2.

3.4.

5. If , then and are parallel.

6.

7.

a b b a

a b c a b a c

0a a a b a b a b 0a b a b

a b c a b c a b c a c b a b c


112/114

The area of parallelogram formed by the two vector anda b a h

where sinh b

Area sina b a b

h

O A

B C

b

a

If the triangle isB


113/114

1

Area of triangle 2 a b

OA

B

(i.e of the parallelogram)1

2

a

b


114/114

THEEND.

GOOD LUCK!!