Lecture 1 Algebra PDF

ALGEBRA M. A. BOATENG, PhD, MIMA KNUST March 8, 2022  TABLE OF CONTENT Sigma Notations Elementary Set Notation Logarithms Exponential Functions Quadratic Functions and Equations M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022...

ALGEBRA M. A. BOATENG, PhD, MIMA KNUST March 8, 2022  TABLE OF CONTENT Sigma Notations Elementary Set Notation Logarithms Exponential Functions Quadratic Functions and Equations M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 2 / 36  SIGMA NOTATION P The symbol (capital sigma) is often used as shorthand notation to indicate the sum of a number of similar terms. For example, Suppose we weigh five children. We will denote their weights by x1 , x2 , x3 , x4 and x5. The sum of their weights x1 + x2 + x3 + x4 + x5 is written more compactly as 5 X xj j=1 So the notation nj=1 xj tells us: P To add the scores xj Where to start: x1 , Where to stop: xn (where n is some number). M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 3 / 36  EXAMPLE 1. Now take the weight of the children to be x1 = 10kg, x2 = 12kg, x3 = 14kg, x4 = 8kg and x5 = 11kg. Then the total weight (in kilograms) is 5 X = x1 + x2 + x3 + x4 + x5 i=1 = 10 + 12 + 14 + 8 + 11 = 55 Notice that we have used i instead of j in the formula above. The j is what we call a dummy variable-any letter can be used. M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 4 / 36  2. Find 4i=1 2xi where x1 = 2, x2 = 3, x3 = −2 and x4 = 1 P Solution P4 i=1 2xi = 2x1 + 2x2 + 2x3 + 2x4 = 2(2) + 2(3) + 2(−2) + 2(1) =4+6−4+2 =8 Exercise 1. Find 6i=1 (xi − 3) where x1 = 2, x2 = 3, x3 = −2, x4 = 12, x5 = −8, P and x6 = 0 P6 2. Find i=1 xi − 8 where x1 = 2, x2 = 3, x3 = −2, x4 = 12, x5 = −8, and x6 = 0 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 5 / 36  Rules of Summation Rule 1: If c is a constant, then n X cxi = cx1 + cx2 +... + cxn i=1 = c(x1 + x2 + x3 +... + xn ) n X =c xi i=1 Rule 2: If c is a constant, then M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 6 / 36  n X c = nc i=1 n X c = c + c + c + c + c + c +... i=1 = nc Rule 3: n X n X n X (xi + yi ) = xi + yi i=1 i= i=1 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 7 / 36  TRIAL Solve the following equations 1. 7i=1 3xi , where i = first seven even numbers P 2. 4j=i=1 (3xi + 2yj ), where j = first four odd numbers and i = first four P prime numbers. 3. 6i=1 (xi − 8)2 where x1 = 2, x2 = 3, x3 = −2, x4 = 12, x5 = −8, and P x6 = 0 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 8 / 36  ELEMENTARY SET THEOREM Definition A SET is a collection of well-defined objects of the same kind. Objects that form a set are called ELEMENTS or MEMBERS of the set. Examples: a. A set of days of the week. b. A set of regions in Ghana. c. A set of counting numbers less than 10 SET NOTATION A set is usually denoted by a capital letter A, B, D, Q, Z, etc, and the members are enclosed in parenthesis {}. Example A={2,4,8} The elements of the set A are 2, 4, 8. M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 9 / 36  OPERATIONS OF A SET Intersection (∩) Two or more sets are said to intersect if there are elements common to the sets in question. For example, If A = {2, 5, 7, 11} and B = {3, 5, 7, 11}. then A ∩ B = {5, 7, 11} Union (∪) When two or more sets are put together to form one set then union of the set is formed. For example If A = {2, 4, 6, 8} and Y = {2, 3, 5, 7}, then A ∪ Y = {2, 3, 4, 5, 6, 7, 8} Complement of a set The complement of a set A is the set of all elements in a universal set U, which are not elements of set A. The complement of A is written as A’ M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 10 / 36  Example Given the universal set K = {countingnumbersfrom1to20}, V = {2, 4, 6, 8, 10, 12} and W = {oddnumbersbetween1and18}. Find a. V ∩ W c. W ′ b. V ∪ W Solution V = {2, 4, 6, 8, 10, 12}, W = {3, 5, 7, 9, 11, 13, 15} a. V ∩ W = {} b. V ∪ W = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15} c. W ′ =??? M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 11 / 36  TRIAL Given that A=even numbers between 1 to 30 B=odd numbers from 1 to 25 D= Prime numbers between 0 and 40 Find A. (A ∩ B) ∪ D B. (A ∩ B) ∩ D c. A′ ∩ D D. B ′ ∪ (A ∩ D ′ ) M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 12 / 36  Properties of Sets Let A, B, and D be three intersecting subsets of the universal set U. We can make the following operational deductions relating the sets A, B, and D Commutative Property A ∩ B = B ∩ A and A ∪ B = B ∪ A, therefore intersection and union are commutative. For example If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8, 10} then A ∩ B = B ∩ A = {2, 4}. Also A ∪ B = B ∪ A = {1, 2, 3, 4, 5, 6, 8, 10} Associative Property (A ∩ B) ∩ D = (A ∩ B) ∩ D and (A ∪ B) ∪ D = A ∪ (B ∪ D) Therefore, intersection (∩) and union (∪) of sets are both associative. For example: If A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and D = {3, 4, 5, 6} then M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 13 / 36  (A ∩ B) ∩ D = (A ∩ B) ∩ D = {4} and (A ∪ B) ∪ D = A ∪ (B ∪ D) = {1, 2, 3, 4, 5, 6, 8} Distributive A ∪ (B ∩ D) = (A ∪ B) ∩ (A ∪ D). Therefore, union of sets (∪) is distributive over intersection of sets (∩). Also A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D). Therefore, intersection of sets (∩) is distributive over union of sets (∪). For example: If A = {1, 2, 3, 4}, B = {1, 3, 5, 7} and D = {2, 3, 5, 7} then A ∪ (B ∩ D) = (A ∪ B) ∩ (A ∪ D). M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 14 / 36  Application of Set Notation There are 20 students in SMS/D I, 16 of them are fluent in French and 10 of them are fluent I English. Each student is fluent in at least one of the two languages. i. Illustrate this information on a Venn diagram ii. How many students are fluent in both English and French? iii. How many students are fluent in only one language? Solution 1. M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 15 / 36  (ii) 16-x + x + 10 -x =20 -x = 20 - 26 -x = -6 x=6 (iii) (16 - x) + (10 - X) = 16 - 6 + 10 - 6 = 14 Trial Mathematics, English and Life Skills books were distributed to 50 students in a class. 22 had Mathematics books, 21 English books and 25 Life Skills books, 7 had Mathematics and English books, 6 Mathematics and Life Skills books and 9 English and Life Skills books. Find the number of students who had (i) All the three books (ii) Exactly 2 of the books (iii) Only Life Skills books M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 16 / 36  LOGARITHM The logarithm of any number b to the base a (written loga b ) is the index or power to which the base must be raised to give the number. From the definition, loga b = c =⇒ b = ac provided a ̸= 1 For example, log2 16 = 4 =⇒ 16 = 24 NB: The base of logarithm can be written in any positive number except 1. The logarithm of a negative number does not exist. When the base of a logarithm is not written, it means it is in base 10 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 17 / 36  LAWS OF LOGARITHMs loga x + loga y = loga (xy ) e.g log2 3 + log2 4 = log2 (12) loga x − loga y = loga ( yx ) eg. log4 21 − log4 7 = log4 ( 21 7 ) = log4 3 loga x n = n loga x eg. log2 52 = 2 log2 5 logb x loga x = logb a log3 8 eg. log2 8 = log3 2 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 18 / 36  PROPERTIES OF LOGARITHMS The following two special properties of logarithms should be noted. Property 1 logb b = 1 i.e. the logarithm of any number to the base of the same number is 1 Property 2 logb 1 = 0 i.e. the logarithm of one (1) to the base of any number other than one is zero (0) M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 19 / 36  TRIAL Use the laws of logarithms to solve the following equations. 1. 2 log2 5 = x 2. log 2 + log x = log12 ‘ 3. log(2x + 3) + log 2 = 1 4. 5 log3 − log3 15 + log3 4 = log3 (3x − 1) M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 20 / 36  EXPONENTIAL EQUATIONS Before we begin with exponential equations, lets have a recap of INDICES PROPERTIES OF INDICES Property 1: (ab)n = an b n eg. (2 × 5)3 = 23 × 53 = 8 × 125 = 1000 Property 2 1 a−n = an M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 21 / 36  NOTE: any number raised to the power -1 is equal to the reciprocal of the number. eg. 2−3 = 213 = 18 Property 3  n a an = b bn 22 eg. ( 72 )2 = 72 = 4 49 Property 4  −n b an = a bn 23 eg ( 23 )−3 = 33 = 8 27 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 22 / 36  Property 5 m √ a n = ( n a)m 2 √ eg. 8 3 = ( 3 8)2 = 4 Property 6 (x n )m = x nm i.e To raise a power to a power, multiply the two powers. eg (33 )2 = 33×2 = 36 = 729 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 23 / 36  EXPONENTIAL EQUATIONS An exponential equation is one in which a variable occurs in the exponent. There are two methods that can be used in solving exponential equations. An exponential equation in which each side can be expressed in terms of the same base can be solved using this property: if b x = b y , then x = y (where b > 0 and b ̸= 1). If the bases are the same, set the exponents to be equal. Examples. 1. 22x = 24 Since they have the same base, we equate the exponents. 2x = 4 =⇒ x = 2 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 24 / 36  TRIAL Solve the following equations 2 +5 1 −2x 2 +4 1 125x = ( 25 ) 2 84x −3 = 322x 1 3 27x = 32x −1 4 93x −2 = 32(2x +1) M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 25 / 36  Solution Method 2: Using logs. 1. Isolate the exponential expression. 2. Take log or ln of both sides, to set up the inverse relationship between exponentials and logarithms. 3. Use this inverse relationship: loga ax = x (where a > 0, a ̸= 1, and loga ax is defined). 4. Solve for the variable. Example 5x = 7 log5x = log7 xlog5 = log7 log7 x= log5 x = 1.209 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 26 / 36  TRIAL 1 15 = 32x +1 2 e x = 43 3 4x +2 − 2 = 12 4 6(2)x +3 = 30 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 27 / 36  QUADRATIC FUNCTIONS AND EQUATIONS A quadratic equation is an equation of the form ax 2 + bx + c = 0 where a, b and c are real numbers and a ̸= 0 There are various ways of solving quadratic equations but we would be focusing on the following: factorization method method of completing squares quadratic formula M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 28 / 36  FACTORISATION METHOD Factoring quadratics is a method of expressing the quadratic equation ax 2 + bx + c = 0 as a product of its linear factors as (x − k)(x − h), where h, k are the roots of the quadratic equation ax 2 + bx + c = 0. Example x 2 + 8x + 12 = 0 x 2 + 6x + 2x + 12 = 0 (x 2 + 6x ) + (2x + 12) = 0 x (x + 6) + 2(x + 6) = 0 (x + 2)(x + 6) = 0 Thus, X1,2 = −2, −6 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 29 / 36  TRIAL Solve the following 4x 2 + 12x + 9 = 0 9x 2 − 4 = 0 2x 2 − 5x + 2 = 0 3x 2 − 8x − 3 = 0 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 30 / 36  COMPLETING THE SQUARES If the coefficient of x 2 is NOT 1, divide both sides of the equation by the coefficient. 2) Isolate all variable terms on one side of the equation. 3) Complete the square (half the coefficient of the x term squared, added to both sides of the equation). 4) Factor the resulting trinomial. 5) Use the square root property Example Solve by completing square y 2 + 6y = −8 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 31 / 36  SOLUTION y 2 + 6y + 9 = −8 + 9 (y + 3)2 = 1 √ y +3=± 1 y = −3 ± 1 y = −4, −2 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 32 / 36  TRIAL Solve the following using the method of completing squaree 1. x 2 − 4x − 8 = 0 2. −4x 2 − 8x − 12 = 0 3. 2s 2 + 5s = 3 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 33 / 36  QUADRATIC FORMULA Given a quadratic equation ax 2 + bx + c = 0 The formula is derived from completing the square of a general quadratic equation. A quadratic equation written in standard form √ −b ± b 2 − 4ac x= 2a Example Solve 11n2 − 9n = 1 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 34 / 36  SOLUTION q 9± (−9)2 − 4(11)(−1) n= 2(11) √ 9 ± 81 + 44 n= 22 √ 9±5 5 n= 22 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 35 / 36  TRIAL Solve the following using the quadratic formula: 1. x 2 − 4x − 8 = 0 2. −4x 2 − 8x − 12 = 0 3. 2s 2 + 5s = 3 M. A. BOATENG, PhD, MIMA (KNUST) ALGEBRA March 8, 2022 36 / 36