MAT 101: Elementary Mathematics 1 Lecture Notes PDF

MAT 101: ELEMENTARY MATHEMATICS 1 LECTURE NOTES BY DR A.S. ONANAYE DEPARTMEMT OF MATHEMATICAL SCIENCES, FACULTY OF NATURAL SCIENCES REDEEMER’S UNIVERSITY, EDE, OSUN STATE, NIGERIA. MAT 101: ELEMENTARY Mathematics I (2 Units) Elementary set theor...

MAT 101: ELEMENTARY MATHEMATICS 1 LECTURE NOTES BY DR A.S. ONANAYE DEPARTMEMT OF MATHEMATICAL SCIENCES, FACULTY OF NATURAL SCIENCES REDEEMER’S UNIVERSITY, EDE, OSUN STATE, NIGERIA. MAT 101: ELEMENTARY Mathematics I (2 Units) Elementary set theory: subsets, union, intersection, complements, Venn diagrams. Real numbers: integers, rational and irrational numbers, remainder theorem and partial fractions, mathematical induction, real sequences and series, theory of quadratic equations, binomial theorem. Complex numbers: Algebra of complex numbers, the Argand diagram, De Morvre’s theorem. Trigonometry: Trigonometric functions of angles of any magnitude, addition and factor formulae.  CHAPTER ONE SET THEORY Definition:Set is a collection of well-defined objects e.g. library or books, kitchen utensils, MTH 101 Students. Set is usually denoted by a capital letter such as ′ , , , … The member, , of a of given set ′ is called an element of the set, ie. ∈. Set Theory:Set theory is a logic of classes—i.e., of collections (finite or infinite) or aggregations of objects of any kind, which are known as the members of the classes in question. Some logicians use the terms “class” and “set” interchangeably; others distinguish between them, defining a set (for example) as a class that is itself a member of some class and defining a proper class as one that is not a member of any class. It is usual to write ∊ for “is a member of” and to abbreviate ∼(x ∊ Y) to implies x ∉ Y. A particular class may be specified either by listing all its members or by stating some condition of membership, in which (latter) case the class consists of all and only those things that satisfy that condition (it is used, for example, when one speaks of the class of inhabitants of London or the class of prime numbers). Clearly, the former method is available only for finite classes and may be very inconvenient even then; the latter, however, is of more general applicability. Two classes that have precisely the same members are regarded as the same class or are said to be identical with each other, even if they are specified by different conditions; i.e., identity of classes is identity of membership, not identity of specifying conditions. This principle is known as the principle of extensionality. A class with no members, such as the class of atheistic popes, is said to be null. Since the membership of all such classes is the same, there is only one null class, which is therefore usually called the null class (or sometimes the empty class); it is symbolized by Λ or or ∅. The notation x = y is used for “x is identical with y,” and ∼(x = y) is usually abbreviated as x ≠ y. The expression x = Λ therefore means that the class x has no members, i.e. it is an empty set, and x ≠ Λ means that x has at least one member or non-empty set. Set Notations and Terminologies Element: The objects or members of a Set = : − ∞ ≤ ≤ ∞. If every element ′ belonging to′ also belongs to ′ ′ i.e. is a Subset of , it is denoted by ⊂ Proper Subset: A Set ′ is said to be proper Subset of Set ′ ′ IF AND IF ONLY IF; * is a Subset of i.e. ⊂ * If there exists at least an element of that is not in or there exists one ∈ such that ∉. Example 2: Let = , = , , , , i.e. Every Set is a Sub-set of itself but proper set. * If is a Set, then ∅ ∪ = where ∅ is a null set or empty set. It is a set that contains no element at all, ie. ∅ = = Λ. Complement of a Set ! Let be a set, then the complement of represented as or is defined: = ∶ ∈ # $% ∉ where # is a referenced set or a universal set. Example 1: Write out the Subset of :- (1) = 1, 2 , they are 1 , 2 , 1,2 , Φ ) Note :- 0 ≠ because 0 is an element ∴ If you have ‘n’ elements in a Set, say , then we should have - = 2. subsets of. Empty / Null and void Set A Set which contains no element is called an empty, null or void Set denoted by Φ , , Λ. Example: A Set of female governors in Nigeria is a null Set. True or False? An even prime number greater than 2 is empty. True or False? A Singleton or unset. E.g. 1 , 2 are singleton sets. A Set which contains only one element is called a Singleton. Example: Even prime number which is 2 and is a singleton. Equality of a Set Two Sets′ ′& ′ are said to be equal if = i.e. ∪ = = Difference of two Sets Let and be two given Sets then − = 0 ∶ ∈ 1 ∉ 2 e.g = 1, 2, 3, , , 4, % = −1, −5, 0, 1, 2, 3, , 4 2 − = ,% 2 − = −1, −5, 0 ∴ − ≠ − ∴ the difference between the Sets is notCommutative, i.e. not equal. Universal Set This contains every element we have in mind at a particular situation. A Set which contains all other Sets as it’s Subsets. It’s denoted by # ) 6. Example: Let = 1, 2, 3, , , 4, % and = −1, −5, 0, 1, 2, 3, , 4 be sets, then # ) 6 = −5, −4, −3, −2, −1, 0, 1, 2, 3, , , 4, %, , 8, 9 is the universal set of set and set. Complement of a Set Let be a Subset of Universal Set # , the Set # , the Set : − = or ! ; = : ∈ < $% ∉ Or using Venn diagram: 6 ; Example.# = 1,2,3,4,5, , , 4, %, = 1, 2, 4, , % = ! = 3, 5, , 4, Exercise 1: Which of the Option is different from the other and why :- (A) Empty, ∧, > ) (B) Null (C) Void (D) Zero Answer is D because Zero is an element Exercise 2: Consider the following Set = , = , 4, , = 4, , ? = 4, ,.@= and >. Question 1: Which of them is a Subset of = , , 4 ? Answer: - , , , ?, @, > are all Subsets. Question 2: Which of them are proper Subsets? Answer: - , , @, > are proper Subsets. Exercise 3: Consider the Set below and answer the question that follows: = 1, = 3, = 4, 1, 3, 5, 2, 9 , ? = 1,2,3,4,5, 9 , @ = 1, 3, 5, 9 # = 1,2 − − − −, 8,9 ,> Insert the correct Symbol ⊂ or⊆between each pair. (i) Φ, ⇒Φ⊂ A (ii) , ⇒ ⊂ (iii) ,@ ⇒ B⊂ @ (iv) ,? ⇒ ⊆? (v) ,@ ⇒ ⊃@ (vi) ? ,@ ⇒?⊃@ (vii) ? ,# ⇒?⊂# Power Set: The Set of all the Subsets of a Set X is called the POWER SET of the Set X. it is usually denoted -0H2. Operation of Set Union of Set: Let and between Sets, the Set that combine both elements of and without repetition is called the UNION OF A AND B. i.e. ∪ = : ∈ ) ∈ ) ∈ ) ∈ $% # ∴ 0 # 2= Example:-  Let = , , 4, %, , 8 = , %, ℎ, J ∴ # = , , 4, %, , 8, ℎ, J = : K =4 = :3 + 2 = 0 ,2 + 3 = 0 ∴ # = M−2, 2, −2N3, −3N2O Remark: -The notion of union of Set can be extended to more than 2 Sets i.e. ∪ ∪ (1) ∪ ∅ = , # = 0 ∪ 2 = 0 ∪ 2-It’s Commutative 0 ∪ 2 ∪ = ∪ 0 ∪ 2–It’s Associative. Intersection Let $% be Sets, then the intersection and is a set that contains all the common elements of and. E.g. ⋃ ∩ = A B Point of Intersection Disjoints Sets:- Set A and B are disjoints i.e. ∩ = ∅ or { } if there’s no point of intersection between the two set i.e. if the 2 Sets don’t have anything in common. 6 A B Two sets are said to be disjoint if They do not have any common Element i.e. ∩ = ∅ ) Symnetic difference of two Sets Let and be two sets, the symnetric difference of and written as ? is ---- i.e. ? = 0 − 2∪0 − 2 A - B := {x : x is in A and x in not in B} Example A = {a, b, c , d} B = {c, d, e ,f} A - B = {a, b} B - A = {e, f} ∴ ? = 0 − 2 ∪ 0 − 2 = , , ,8 , 66 A B A B 0 − 20 − 2 Order of a Set The Order of a finite Set is the number of element in that Set. For example:- = 1, 2, 4 ∴$0 2=3 U−A=A Example 1:- Prove that − = ∩ ; by (i) Using Venn diagram ,(ii) By rigorous method 6 6 A B A B 0 − 2 ; 6 A B $ ; 2. ∴ − ⊆ $ ; $ ; ⊆ − Let belong to − i.e. S − = ∈ $% ∉ = ∈ $% ∈ Let belong to $ i.e. ∈ and ∈ ∈ and ∉ i.e. ∈ − Hence − = $ OR A = {a, b, c , d} B = {c, d, e ,f} A - B = {a, b} = , , ∩ = , , ∴ − = ∩ = ,.  Exercise: Prove that 0 : 21 = $ Using (i) Venn diagrams (ii) Rigorous method 66 A B A B 0 ∪ 20 ∪ 21 6 A BA B ; ; Let belong to 0 : 2′ ∈ and ∈ ∈ $ , Let belong to $ ∴ ∈ and ∈ ∉ or ∉ ∈ 0 : 21 * Assignment * Prove rigorously :0 ∩ 2= 0 : 2∩ 0 : 2 APPLICATION OF SET THEORY Example 1:- In Maths 101 tutorial group of 30 Students, 17 Students Studied Physical Sciences; 15 Mathematical Sciences. While 10 Studied neither Mathematical nor Physical. (i) How many Students Studied both? (ii) Studied Mathematical Science only? (iii) Studied Physical Sciences only? Solution + = 17 Phy Sci =17 Math Sci=15 = 17 − + = 15017 − 2 015 − 2 ∴ = 15 − ↓↓ 10 17 − + + 15 − + 10 = 30 32 + 10 − = 30 ∴ 12 Students Studied both. (ii) For Mathematical Science only = 15 − = = 15 − 12 ∴ =3 ∴ 3 Students Studied Mathematical Science only. (iii) For Physical Science only = 17 − = = 17 − 12 ∴ =5 ∴ 5 Students Studied Physical Science only. Example 2:- Let , , be Subsets of a universal Set # defined by ⨂ = −0 ∩ 2 Draw on a separate Venn Diagrams. (i) 0 ; ⨂ 2⨂ ; (ii) 0 ⨂ 2⨂ (Assignment) Solution 0 ;H 2H ; Assumption: - Let 0 ; H 2be ? ?H ; = ? − 0? $ ;2 ⇒ ; H =?= ; −0 ; $ 2 ⇒ W ; − 0 ; $ 2X H ; (i) ?H ; = ? − 0? $ ; 2 ⇒ W ; − 0 ; $ 2X − YZ ; − W ; $ X[$ ; \ (1) (2) A B A B C C (3) A B  C ; $ (4) (5) A B A B C C ; −0 ; $ 2 ; (6) A B C 0 ; $ 2$ ; (7) A B C  ; − ]0 ; $ 2$ ; ^ (8) A B C C Z ; −0 ; $ 2 − W0 ; − W0 ; $ 2X2$ ; X[ Exercise 3:- Suppose that 100 of the 120 Mathematical Students of the FNS take at least 1 of the major Nigerian Languages. Suppose 65 Hausa, 45 Igbo, 42 Yoruba. 20 Studied Hausa and Igbo, 25 Studied Hausa and Yoruba, 15 Studied Igbo and Yoruba. Find (1) Number of Students who studied all languages (2) Fill in the correct number of Students in each of the 8 region on the Venn diagram. Example 4:- Let , , be Subsets of Universal Set # defined H = ; ? ; Draw on a separate diagram REAL NUMBERS Integers are the numbers between – Infinity to Infinity and it’s denoted by _ _= : −∞ ≤ ≤∞ Natural numbers are the positive integers between 1 to Infinity and it’s denoted by ` `= :1 ≤ ≤∞ a Rational numbers are numbers that can be written as bprovided′ ′is not equal to 0. It’s represented as Q a Q =M b , ≠ 0O a Irrational numbers )numbers that cannot be written as b e.g √3 ,∧ Real Number: A real number is a number on the real line SURD: They )Irrational numbers. Surd is the root of an arithmetic number whose value can’t be obtained exactly e.g√2, 2 √3 Different Surd can be combined together by arithmetic operation and this is called a SURD OPERATION A Surd that comprises of a single term is called monomial Surd e.g. 2√3 RATIONALIZATION OF SURDS CONJUGATE SURD Suppose we have a binomial Surd of the form )′′ √3 + √2 ′′then Conjugate of the Surd is √3 − √2. Rationalizationof Surd is a means of simplifying a Surd expression in its Lowest or Simplest form. Example 1:- 3√2 − 5√3 d 1 $ ef √3 − √2 Solution 3√2 − 5√3 H √3 + √2 ⇒ √3 − √2 √3 + √2 3√2 − 5√3 √3 + √2 ⇒ ]√3^ ]√2^  3√6 + 6 − 5.3 − 5√6 ⇒ 3−2 3√6 − 5√6 + 6 − 15 ⇒ 1 ⇒ −2√6 − 9 ⇒ −]2√6 + 9^ ⇒ −]9 + 2√6^ Example 2:- Rationalize 4√5 + 6√3 √5 − √3 Solution:- 4√5 + 6√3 H √5 + √3 ⇒ √5 − √3 √5 + √3 SQUARE ROOTS OF SURDS Suppose we have a Surd expressed as:- h + √ and the Square root is required, the procedure involved assuming:- i + √ = √ + hj Squaring both sides of the equation kh + √ l2= ]√ + hj^2 +√ = +j +2h j ∴ +j = −0 2 √ = 2h j − 0 2 Example 1:- Find the Square root of 7 + 2√6 h7 + 2√6 = √ + √ Square both sides Z]7 + 2√6^ 1N2[2 = ]√ +√ ^ 7 + 2√6 = + + 2√ * ∴7= + − 0H2 2√6 = 2 √ ∴ √6 = √ − 0 2 *∴ 6= − 0Hm 2 From equation 0H2 , = 7 − − 0H mm 2 Substitute 07 − 2 for a in equation 0Hm2 6=7 − K K −7 +6=0 0 − 620 − 12 = 0 ∴ =6 )1 =7− , =6 =7−6 ∴ =1 When =1 =7−1 =6 ∴ The Square root of 7 + 2√6 = √1 + √6 or √6 + √1 Exercise 5:- 3√5 − 5√2 d 1 $ ef 02 4 − 3√5 3√2 − 5√2 0 2 √3 + √2 Find the square root of the following:- 0 25 − 2√6 0 24 + 2√5  0 27 − 2√6 REMAINDER THEOREM It states that if a polynomial 8 0 2 is divided by 0 − 2 , the quotient would be a polynomial 90 2 of one degree less than the degree of 8 0 2 together with a remainder ′d still to be divided by 0 − 2. 80 2 d = 90 2 + − − 80 2 = 90 20 − 2 + d When = , 80 2 = 90 20 − 2 +d 80 2 = d If 80 2 is equal to d and is equal to zero, then0 − 2 is a factor of 80 2 and the remainder theorem is now a factor theorem. Procedure involving the use of remainder theorem The step Involves the trial and error method were the polynomial are tested for various values that will give a remainder Zero. Example 1:- Factorize and find the root of the equation:- 8 012 = n + 5 K − 2 − 24 = 0 8 012 = −1 + 5 − 2 − 24 ≠ 0 ∴ 8 0 12 ≠ 0 8012 ≠ 0 When = 2 ∴ 0 − 22 is a factor K + 7 + 12 n + 5 − 2 − 24 K x-20 n − 2 K 2_____________ 7 K−2 −0 − 0+7 K − 14 2 +12 − 24 −012 − 242 0 0  n +5 K − 2 − 24 = 0 − 220 K + 7 + 122 n +5 K − 2 − 24 = 0 − 220 + 320 + 42 = 0 e $ ) 8 41 )o ∴ 2 , 3 , −4 ) roots of the equation and 0 − 22 , 0 + 32 and 0 + 42 are factors Example 2:- (i) 5pn + 31pK + 31p + 5 = 0 , find the roots of the equation 80 2 = (ii) Express the function q 2 r − 7 n − 2 K + 13 + 6 (a) As (b) If -0 2 = 0 − 220 K − − 22 explain why -022 = 0 Answer: It’s because 0 − 22 is a factor. PARTIAL FRACTION We can add fractions together to produce a single fraction. This single fraction can be simplified back to its original form. This simplified fraction is called a PARTIAL FRACTION of the ones Simplified. e.g. 1 1 2 + = 1− 1+ 1− K 1 1 2 + = −1 +1 K −1 There are different forms of partial fraction (i) The one with each denominators is linear (ii) Denominators can’t be Simplified (iii) Repeated denominators (iv) Improper fractions Theare different methods of resolving these fraction: They are:- (i) Cover up method (ii) Equating Coefficient method (i) Linear denominator: This is anequation of the form + = + 0 − 4 20 + %2 −4 0 + %2 Example 1:- +7 d o es K − 7 + 10 +7 = + 0 − 520 − 22 0 − 52 0 − 22 Multiply each side by 0 − 520 − 22 + 7 = 0 − 22 + 0 − 52 When =2 9 = 0 2 − 22 + 0 2 − 52 9 = −3 ∴ = −3 When = 5  5 + 7 = 05 − 22 + 05 − 52 12 = 3 + 0 12 ∴ = 3 ∴ =4 +7 4 −3 ∴ = + 0 − 520 − 22 0 − 52 0 − 22 +7 4 −3 ∴ = + 0 − 520 − 22 0 − 52 0 − 22 +7 4 3 ∴ = − K − 7 + 10 0 − 52 0 − 22 Alternative +7 = + 0 − 520 − 22 0 − 52 0 − 22 Multiply each side by 0 − 520 − 22 + 7 = 0 − 22 + 0 − 52 +7= −2 + −5 +7= + −5 −2 +7= 0 + 2−5 −2 Equating the Coefficient of + =1−0 2 Equating the Coefficient of the Constant 7 = −5 − 2 _______0 2 From equation 0 2 = 1 − ______0 2 Substitute 01 − 2 for in equation 0 2 7 = −501 − 2 − 2 7= 5+5 −2 12 = 3 ∴ = 12N3 ∴ =4 =1− , =4 ∴ =1−4 ∴ = −3 +7 4 3 ∴ = − 0 − 520 − 22 0 − 52 0 − 22 Denominator not Simplified: - au vb zuvb This is always of the form u w vxuvy = {u w vxu Where the denominator can’t be simplified Example:- |u w v }u~; Resolve 0Kuvn20uw v|uvK2 Solution:- 5 K +9 −1 + + = + 02 + 320 + 5 + 22 02 + 32 0 K K + 5 + 22 Multiply each side by 02 + 320 K + 5 + 22 5 K + 9 − 1 = 0 K + 5 + 22 + 02 + 320 + 2 5 K +9 −1= K +5 +2 +2 K +2 +3 +3 5 K +9 −1= K +5 +2 +2 K +2 +3 +3 5 K +9 −1= K +2 K +5 + +2 +3 +2 + K0 5 K +9 −1= + 2 2 + 05 + 2 + 3 2 + 2 + 3 K Equating the Coefficient of 5 = + 2 ______0 2 Equating the Coefficient of 0 2 _______________ 9 = 5 + 2 + 3 ____________0 2 Equating the Constant −1 = 2 + 3 ________0 2 From equation 0 2 = 5 − 2 ________0 s 2 Substitute 05 − 2 2for in equation 0 2 PARTIAL FRACTION (CONTINUED) Multiply equation 0 2 by 2 5= +2 ×2 10 = 2 + 4 _______0 s2 From equation 0 2 = 5 − 2 ________0 s2 Substitute equation 0 s 2 into 0 2&0 2 9 = 505 − 2 2 + 2 + 3 9 = 25 − 10 + 2 + 3 9 = 25 + 2 − 7 3× −16 = 2 − 7 ________0×2 −1 = 205 − 2 2 + 3 −1 = 10 − 4 + 3 2 × −11 = 3 − 4 ___________0××2 −48 = 6 − 21 _______0×××2 −22 = 6 − 8 ________0××××2 Subtracting equation 0×××2_______0××××2 −48 − 0−222 = −21 − 0−8 2 22 − 48 = 8 − 21 −26 = −13 ∴ =2 16 = 2 − 7 , = 2 16 = 2 − 7022 −16 = 2 − 14 −20 = 2 − 2 = +2 −2 =.. −2N2 = −1 2 = +1 5= +2 , =2 PARTIAL FRACTION (CONTINUED) 5 K+9 −1 1 2 +1 ∴ = + 02 + 320 + 5 + 22 02 + 32 0 K K + 5 + 22 (iii) Repeated denominator:- auvb z € This is of the form 0{uvx2 = 0{uvx2 + 0{uvx2K f −−− 0 + % 2$ Example 1:- ;ru w vn;uv| Resolve 0u~;20Kuvn2K Solution:- 14 K + 31 + 5 ⇒ + + 0 − 1202 + 322 0 − 12 02 + 32 02 + 322 Multiply each side by 0 − 1202 + 322 14 K + 31 + 5 = 02 + 322 + 0 − 1202 + 32 + 0 − 12 When = 1 14012 2 + 31012 + 5 = 02012 + 322 + 01 − 1202012 + 32 + 01 − 12 14 + 31 + 5 = 25 + 0 +0 50 = 25 50 ∴ = 25 ∴ =2 PARTIAL FRACTION (CONTINUED) + 5 = 0022+ k−5N2l002 + k−5N2l } } 14 × r − K 126 93 5 −5 − + =0 + 0 4 2 1 2 126 − 186 + 20 −5 = 4 2 −60 + 20 −5 = 4 2 −40 −5 = 4 2 −80 = −20 ∴ =4 ∴ =3 When = −1 140−122+30−12 + 5 = 2020−12 + 322+ 0−22012 + − 2042 14 − 31 + 5 = 2 − 2 − 8 −12 = −6 − 2 −12 = −6 − 2 −12 + 6 = −2 −6 = −2 ∴ = −6N−2 ∴ =3 14 K + 31 + 5 2 3 4 ∴ = + + 0 − 1202 + 322 0 − 12 02 + 32 02 + 322 (iv) Improper fraction:- It’s when the degree of the numerator is greater than the degree of the denominator. Example:- Resolve into partial fraction L 3 K ⇒ −5 +4 3 K − 5 + 43 K