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Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Definition : A set is well- defined collection of objects. The objects in sets can be
anything . These objects are called the elements or members of the set .
Empty set or null set : is the set without any elements , denoted by
 .
For example : Let X be any set. Then   x  X / x  x  .
We say that a set A is a subset of set B if every element of A is also an element of B and we write
that A  B .
The intersection of sets A and B , denoted by A  B , is the set of all elements belonging to both
set A and set B , i.e.
A  B  x / x  A and x  B  .
The Union of sets A and B , denoted by A  B , is the set of all elements belonging to set A, to set
B ,or to both, i.e.
A  B  x / x  A or x  B  .
SET OF NUMBERS
I.
P
= {1,2,3,….} = the set of Natural numbers, or Positive Integers.
II.
= { 0,1,2,…} = the set of non negative Integers .
III.
= { …,-3,-2,-1,0,1,2,3,…} = the set of Integers .
IV.


a
b
/ a  ,b 

and b  0 = the set of rational numbers .
V. j = { x \ x can’t be expressed as a rate of tow integers } = the set of irrational
numbers .
VI.
= { all rational and irrational numbers } = the set of real numbers .
Real number
rational numbers
irrational numbers
Integers
Negative Integers
Zero
Positive Integers
1
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Definition : A Prime number is a positive integer other than 1 that has no positive integer
factors other than itself and 1 .
Definition : A Composite number is a positive Integer greater than 1 that is not a prime
number .
Example (1)
Determine which of the following numbers are
(a) Integers
(b) Rational numbers (c) Irritation numbers
(d) Real numbers
(e) Prime numbers
(f) Composite numbers
-0.3 , 0 , 6 , 7 , 41 , 51 , 0.717717771000 ,  ,
2 , 5
Solution :
(a) Integers
: 0 , 6 , 7 , 41 , 51
(b) Rational numbers
: -0.3 , 0 , 6 , 7 , 41 , 51 , 5
(c) Irritation number
: 0.717717771… ,  ,
(d) Real numbers
: -0.3 , 0 , 6 , 7 , 41 , 51 , 0.717717771… ,  ,
(e) Prime numbers
: 7 , 41
2
2 , 5
(f) Composite numbers : 6 , 51
Example (2)
Given A = {0,1,4,6,9} , B = {1,3,5,7,9} and C = {x \ x is a Prime number < 10} .
Find (a) A ∩ B
(b) A ∩ C
(c) A  B
(d) A C
Solution :
(a) A ∩ B = {1 , 9}
(b) A ∩ C =   , note that C = {2,3,5,7}
(c) A  B = {0,1,3,4,5,6,7,9}
(d) A C = {0,1,2,3,4,5,6,7,9}
2
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Properties of Fractions :Let a and b real numbers , where b ≠ 0 and d ≠ 0 , then equality :
Equivalent fractions :
Addition :
a
c
a c
 
b
b
b
Multiplication :
Division :
Sign : 
a
ac
,c≠0

b bc
a
c
a c
 
b
b
b
Subtraction :
a
b
a c
if and only if a d = b c

b d
,
a
c
ad b c


b
d
bd
,
a
c
ad b c


b
d
bd
c
a c

b bd
a
c
a


b
d
b
d
ad

c
bc
,c  0
a
a
a


b
b
b
Division properties of Zero :
1. for a ≠ 0 ,
2.
0
 0 ( zero divided by any non zero number is zero )
a
0
is undefined ( Division by zero is undefined )
a
Example (1)
Perform the indicated operations, where a is a nonzero real number .
2a
3a
5 2
12 a
2 a
(a)
(b)
(c)


5
7
7
3
5
3
(d)
2
2
 3
5
3
Solution :
(a)
2a
3 a 14 a  15 a 29 a



5
7
35
35
5 2  10

7
3
21
12 a
2 a 12 a


(c)
5
3
5
(b)
3
36 a
18

 
2 a
10 a
5
3
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
2
2 2 11
2
 3  

5
3 5
3
5
(d)
3
6

11
55
Natural number exponents :
(1) if b is any real number and n is any natural number , bn = b.b.b.b…..b (n factors of
b ) where b is the base , n is the exponent , and bn is the nth power of b .
1
1
(2) b-n =
and  n
= bn , where b ≠ 0
n
b
b
(3) b0 = 1 , where b ≠ 0
Properties of Exponents :
If n , m and p are integers and a and b are real numbers, then
1. bn . bm = bn+m ( product )
2.
= bn-m , b ≠ 0 ( quotient )
n m
nm
n
3. (b ) = b
m p
np
mp
,(a b ) =a b
 an 
,  m 
b 
p

a np
b ≠ 0 ( power )
b mp
Properties of Redicals :
If n and m are natural numbers and a and b are nonnegative real numbers, then
1.
2.
3.
n
a
n
a
n
n

b
m n
a
b 
n

n
( product )
ab
a
b
mn
(Quotient )
a
(Index )
Example :
 2abc 2 
(a) 

2
 5a b 
Simplify
3
(b)
x n y 2n
x n 1 y n
(b)
3
162x 4 y 6
Solution :
3
 2abc 2 
(a) 
 =
2
 5a b 
 2c 2 


 5a 
3
=
8c6
125 a 3
4
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
x n y 2n
=
x n 1 y n
(b)
(c )
3
4
6
162x y =
xn-(n-1) y2n-n = x yn

1
1
1
4 6 3
3
3
2 3 3
2
2
162x y
 3 . 6x . x . ( y )
 3 xy (6x ) 3  3xy 3 6x
 

Polynomials :
The general form of a polynomial of degree n in the variable x is :
anxn + an-1 xn-1 + … + a2 x2 + a1 x1 + a0,
Where an ≠ 0 and n is nonnegative integer . the coefficient an is the leading coefficient , and
a ≠ 0 is the constant term.
Example (1)
Simplify ( 3x – 4 ) ( 2x2 + 5 x + 1 )
Solution :
( 3x – 4 ) ( 2x2 + 5 x + 1 ) = (3x – 4 ) ( 2x2 ) + ( 3x – 4 ) ( 5x ) + ( 3x – 4 ) ( 1 )
= 6x3 – 8x2 + 15 – 20x + 3x – 4
= 6x3 + 7x2 – 17x – 4
Factoring :
Definition
:
Uniting a polynomial as a product of polynomials of lower degree is called Factoring .
Factoring Formulas :Difference of toe sequences : x2 + y2 = ( x + y ) ( x – y )
Prefect – square trinomials :
x2 + 2xy + y2 = (x + y )2
x2 – 2xy + y2 = ( x – y )2
Sum of cubes
:
x3 + y3 = ( x + y ) ( x2 – xy + y2 )
Difference of cubes
:
x3 – y3 = ( x – y ) ( x2 + xy + y2 )
Examples : Factor
(a) 49x2 – 144 (b) 16x – 40xy + 25y2
(e) x2+10xy + 25y2 – z2
5
(c) 8x3 + y3
(d) x3 – 64
(f) x2 + 7x + 12
(g) 4x4 – 25x2 + 36s
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Solution :
(a) 49x2 – 144 = (7x)2 – (12)2 = (7x + 12 ) ( 7x – 12 )
(b) 16x2 – 40xy + 25y2 = (2x)2 – 2(4x)(5y) = (4x – 5y )2
(c) 8x3 + y3 = (2x)3 + y3 = ( 2x+y ) ( 4x2 – 2xy + y2 )
(d) x3 – 4 = x3 – (4)3 = ( x - 4 ) ( x2 + 4x + 14 )
(e) x2 + 10xy + 25y2 – z2 = ( x2+ 10xy + 25y2) – z2 = ( x + 5y )2 – z2 = ( x+ 5y + z )( x +
5y – z)
(f) x2 + 7x + 12 = ( x + 3 ) ( x + 4 )
(g) 4x2 – 25x2 + 36 = ( 4x2 – 9 ) ( x2 – 4 )
Equations :
Definition :
A linear equation in the single variable x is an equation that can be written in the form
ax + b = 0
Where a and b are real numbers, with a ≠ 0
Example : Solve 2 x  10  x  36
5 5
3
Solution :
2 x  10  x  36
5 5
3
2 x  x  36  10
5 5
3
7 x   14
5
15
x   14  7   14
5
5
15
15   6
7
Absolute Value Equation :
For any variable expression A and any nonnegative real number k,
A
=k
if and only if
A = k or A = -k.
6
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Example : Solve 2x  5  21
Solution :
2x  5  21 implies that 2x – 5 = 21 or 2x – 5 = - 21. Solving these equations
2x – 5 = 21
2x – 5 = - 21
or
2x = 26
2x = -16
x = 13
x=-8
therefore, the solutions are -8 and 13 .
Quadratic Equation :-
Definition :
A Quadratic Equation in x is an equation that can be written in the standard quadratic form
ax2 + bx + c = c , a ≠ 0.
Definition ( Zero product property ) :If A and B are algebraic expressions, then
AB = 0 if and only if A = 0 or B = 0.
Definition of a Complex number :
If a and b are real numbers and i is the imaginary unit, then a + bi is called a complex
number. The real number a is called the real part and the real b is called the imaginary
part of the complex number .
Definition ( the quadratic form ) :If ax2 + bx + c = 0 , a ≠ 0 , then
x 
b 
b2 4a c
2a
Example : Solve the fallowing equations :
(a) 3x2 + 10 x = 8
(b) (x + 1 )2 = 49
(c) x2 = 3x – 5
solution :
(a) 3x2 + 10 x = 8
7
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
3x2 + 10 x – 8 = 0
(3x – 2 ) ( x + 4 ) = 0
3x – 2 = 0
or
x+4=0
2
3
or
x = -4
x=
therefore, the solutions are - 4 and
2
3
(b) (x + 1 )2 = 49
x+1= ±7
thus x = - 1 + 7 = 6
x = -1 – 7 = - 8
or
the solutions are – 8 and 6 .
(c) x2 = 3x – 5
x2 – 3x + 5 = 0
a = 1 , b = -3 , c = 5
then;
x 
therefore,
b 
b2  4a c
2a
x
3  i

3
9  20
2

3
 11
2
11
2
the solutions are
x
3  i
2
11
and
x
3  i
11
2
Functions :
Definition ( Relation ) :
Let A and B are given sets. A relation from A to B is a subset of A x B.
Example :
Let A = { 1,2 } , B = { 3,4 }. Then the subset R = { (1,3) , (2,3) } is a relation from A to B
Definition ( Function ) :A function f from a set A into a set B is a rule which assigns to a each element of a set A
a unique element in a set B .
8
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Function :
If f is a function from A to B, then the set A is called the domain of the function f and the
range of f is the set
rang (f) =  b  B
/  a  A with b  f (a )

Example (1) :
Decide which are function and find their domain and rang. Let A = {a,b,c}, B = {b,c,d ,e }
(a) f = { ( a,b ) , (a,c) , (b,d) , (c,e) }.
(b) g = { (a,c) , (b,d) , (c,e) }.
(c) h = { (a,b) , c,d} } .
solution :
(a) f is not a function from A to B, since f(a) = b and f(a) = c .
(b) g is a function from A to B .
Domain (g) = { a , b , c }
Rang (g)
={c,d,e}
(c) h is not a function from A to B , since b  A is not assigned to any element of
B.
Example (2)
Which relations define y as a function of x ?
(a) 3x + y = 1
(b) y2 – 4 x2 = 9
Solution :
(a) solving 3x + y = 1 for y yields y = - 3x + 1 . since -3x + 1 is unique real
number for each x, then this equation defines y as a function of x .
(b) solving y2 – 4x2 = 9 for y yield y  
4 x 2  9 , since 
4 x2  9
produces two values of y each value of x . For example when x = 0, then y = 3 or
y = - 3. Then this equation does not define y as a function of x .
Definition ( Identity Functions ) :Let A be any set . Let the function f : A A be defined by the formula
f(x) = x , for all x  A
that is, f assign to each element in A that element itself.
9
Dr. Falleh Al-Solamy
Arab Open University
‫الجامعة العربية المفتوحة‬
‫المملكة العربية السعودية‬
Kingdom of Saudi Arabia
Then f is called the Identity Function .
Definition ( Constant Function ) :A function f of A into B is called a constant function if the range of f consists only one
element.
For example, let f(x) = 5 for all x 
then f is a constant function .
Properties of Functions :(1) One – to – one Function ( or Injective ) :Let f : A  B be a function . then f is called One – to – One or Injective if every a , b  A,
f (a) = f (b) implies a = b
or
a ≠ b implies f(a ) ≠ f(b)
(2) Onto Function ( Sirjective ) :Let f : A  B be a function . then f is called onto or sirjective if for any element b  B
there exists an element a  A, such that f (a) = b .
In other words, if range(f) = B, then f is called onto function .
(3) Bijective Function :Let f : A  B be a function . if f is both injective and surjective, that is , both One – to –
One and onto , then f is called a bijective function .
10
Dr. Falleh Al-Solamy