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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