Form 2 Mathematics – SETS msomimaktaba, November 13, 2018August 17, 2024 SETSA set is a group/ collection of things such as a herd of cattle, a pile of books, a collection of trees, a shampoos bees and a fleck of sheepDescription of sets-A set is described/denoted by Carl brackets { } and named by Capital lettersExamplesIf A is a set of books in the library then A is written asA= {All books in the library} and read as A is a set of all books written in the library-The things/objects In the set are called Elements or members of the setExampleIf John is a student of class B, then John is a member of class B and shortly denoted as £B.If A= {1,2,3} then 1A, 2A and 3A}The number of elements in a set is denoted by n(A)ExampleIf A= {a, e, i, o, u} then n (A) =5ExampleIf A is a set of even, describe this set bya) Wordsb) Listingc) FormulaSolution:a) By words;A= {even numbers}b) By ListingA= {2, 4, 6, 8…}c) By FormularA= {x: x= 2n} where n= {1, 2, 3…} and is read as A is a set of all element x such that x is an even number.Describe the following sets by ListingA= {whole numbers between 1 and 8}Solution:A= {2, 3, 4, 5, 6, 7}Write the following sets in wordsA= {an integer < 10} Solution:A= {integers less than ten} or A is a set of integers less than tenTYPES OF SETSFinite set; Is a set where all elements can be counted exhaustively.Infinite set: An Infinite set is a set that all of its elements cannot be exhaustively countedExampleB= {2, 4, 6, 8…}An Empty set; Is a set with no elements. An Empty set is denoted by { } or ØExampleIf A is an Empty set then can be denoted as A= { } or A= ØExerciseList the elements of the named setsA= {x: x is an odd number < 10}A= {1, 3, 5, 7, 9}B= {days of the week which began with letter S}B= {Saturday, Sunday}C= {Prime numbers less than 13}C= [2, 3, 5, 7, 11}Write the named sets in wordsB= {x: x is an odd number < 12}B is a set of x such that x is an odd number less than twelveE= {x: x is a student in your class}E is a set of x such that x is a student in your classWrite the named sets using the formula methodsA= {all men in Tanzania}A= {x: x is all men in Tanzania}B= {all teachers in your school}B= {x: x is all teachers in your school}C= {all regional capital in Tanzania}C= {x: x is all regional capital in Tanzania}D= {b, c, d, f, g…}D= [x: x is a consonant}COMPARISON OF SETS-SET may be equivalent, equal or one to be a subset of other-Equivalent sets are sets whose members (numbers) match exactlyExampleA= {2, 4, 6, 8} and B= {a, b, c, d}Then A and B are equivalentThe two sets can be matched asGenerally if n(A) = n(B) then A and B are equivalent setsExampleIf A= {1, 2, 3, 4} and B= {1, 2, 3, 4} since n(A) = n(B) and the elements are alike then set A is equal to set BSubset: Given two sets A and B, B is said to be a subset of A. If all elements of B belongs to AExampleIf A= {a, b, c, d, e} and B= {a, b, c, e}, Set B is a subset of A since all elements of set B belongs to set A. But set B has less elements than set A. Then set B is a proper subset of set A and A is a super set of B.If A= B then either A is an improper subset of B or B is an improper subset of A.Symbolically written as A⊆B or B⊆ANote: an empty set is a subset of any set-The number of subset in a set is found by the formula 2n where n= number of elements of a setExampleList all subset of A= {a, b}Solution:2n, n = number of element in a set. So 22 = 4The number of subset = 4The subset of A are { }, {a}, {b}, {a, b}How many subset are there in A= {1, 2, 3, 4}Solution:The number of subset= 24= 16UNIVERSAL SET [U]-Is a single sets which contains all elements sets under consideration for example the set of integers contains all the elements of sets such as odd numbers, even numbers, counting numbers, and whole numbers. In this case the set of integers is the Universal set.Exercise1. Which of the following sets area) Finite setb) Infinite setc) Empty setA= {Nairobi, Dar es Salaam}B= {2, 4, 6…36}E= {All mango trees in the world}F= {x: x is all students aged 100 years in your school}H= {1, 3, 5, 7}D= {all lions in your school}I= ØSolution:a) Finite set areA= {Nairobi, Dar es Salaam}B= {2, 4, 6…36}H= {1, 3, 5, 7}Infinite setE= {All mango trees in the world}F= {x: x is all students aged 100 years in your school}b) Empty sets areD= {all lions in your school}I= Ø2. In each of the following pairs of sets shown by matching whether the pairs are equivalent or not Equivalent are:A= {a, b, c, d} and B= {b, c, d, e}Which are not equivalent are:B= {Rufiji, Ruaha, Malagarasi} and C= {lion, leopard}B and C are not equivalent.Which of the following sets are equalA= {a, b, c, d}, B= {d, a, b, c}, C= {a, e, I, o, u}, D= {a, b, c, d}, E= {d, c, b, a}, F= {a, e, b, c, d}SolutionA, B, D and E are equalList all subsets of each of the following setsa) A= 1The number of subset = 21= 2Therefore; The subset of A are { }, {1}b) B= ØTherefore; number of subsets is { }c) C = {Tito, Juma}Number of subset = 22=4Therefore; the subsets of C are { }, {Tito}, {Juma}, {Tito, Juma}Name the subsets of each pair by using the symbol ⊂a) A= {a, b, c, d, e, f, g, h} and B= {d, e, f}Therefore = B⊂Ab) A= {2, 4} and D= {2, 4, 5} = A⊂Dc) A= {1, 2, 3, 4 …} and B= {2, 4, 6, 8…} = A⊆BGiven G = {cities, towns and regions of Tanzania} which of the following sets are the subsets of G?A= {Nairobi, Dar es Salaam}B= {Dodoma, Mombasa, Mwanza}C= { }D= {Arusha, Iringa, Bagamoyo}E= {Mbeya, Tunduru, Ruvuma}Therefore; the subsets of G are C, D, EWhich of the following sets are the subsets of K given that K= {p, q, r, s, t, u, v, w}A= {p, s, t, x}B= {q, r, d, t}C= { }D= {p, q, r, s, t, u, v, w}E= {a, b, c, d}F= {s, v, q}Therefore; the subsets of K is D, C, FWhat is n(A) if A= { }n(A) = 0Write in words the universal set of the following setsa) A= {a, b, c, d}The universal set of A is a set of alphabetsb) B= {1, 2, 3, 4}The universal set of set B is the set of natural numbersOPERATION WITH SETSUNIONThe union of two sets A and B is the one which is formed when the members of two sets are putted together without a repetition. Thus the union is , this union of A and B can be denoted as AB is defined as x; XA or XBExampleIf A= {2, 4, 6} and B= {2, 3, 5} then AB= {2, 4, 6} {2, 3, 5}= {2, 3, 4,5, 6}Find AB when A= {a, b, c, d, e, f} and B= {a, e, I, o, u}Solution:AB= {a, b, c, d, e, f, I, o, u}INTERSECTIONThe Intersection of two sets A and B is a new set formed by taking common elements. The symbol for intersection is “”ExampleA= {1, 2, 3, 4, 5}, B= {1, 3, 5} then AB= {1,3,5}Find AB if A= {a, e, i, o, u}, B= {a, b, c, d, e, f} then AB= {a, e}COMPLEMENT OF A SETIf A is a subset of a universal set, then the members of the universal set which are not in A, form compliment of A denoted by A΄ExampleIf = {a, b, c … z} and A= {a, b} then A΄= {c, d, e, … z}Given that U= {15, 45, 135, 275} and A= {15} find A΄ Solution:A’= {45, 135, 275}JOINT AND DISJOINT SETSJOINT SETS; Are sets with common elementsE.g. A= {1, 2, 3, 5}, D= {1, 2} then A and D are joint sets since {1, 2} are common elementsDIS JOINT SETS; Are sets with no elements in commonFor example A= {a, b, c} and B= {1, 2, 3, 4} then A and B are disjoint sets since they do not have a common elementEXERCISE1.Finda) Unionb) Intersection of the named setsA= {5, 10, 15}, B= {15, 20}a) AB = {5, 10, 15, 20}b) AB= {15}A= { }, B= {14, 16}a) AB= { , 14, 16}b) AB= { }A= {First five letters of the English alphabet}, B= {a, b, c, d, e}a) AB= {a, b, c, d, e}b) AB= {a, b, c, d, e}A= {counting numbers}, B= {prime numbers}a) AB= {counting numbers}b) AB= {prime numbers}A = {o, }, B= { }VENN DIAGRAM-Are the diagrams (ovals) devised by John Venn for representation of setsExampleIf A= {a, b, c} can be represented asµ is the universal set, in this case is the set of all English alphabets. If the set have any elements in common, the ovals over lap for example, If A= {a, b, c} and B= {a, b, c, d} then it can be represented as Disjoint sets also can be represented on a Venn diagramExample: If A= {a, b}, B= {1, 2} the relation A and B is as follows ExamplesIf A is a subset of B, represent the two sets on a Venn diagramRepresent A= {2, 3, 5}, B= {2, 5, 7} C= {2, 3, 7} in a Venn diagramSolution: Represent AUB in a Venn diagram given that A= {1, 2}, B= {1, 3, 5}Solution:If set A and B have same elements in common, represent the following in a Venn diagrama) ABb) ABSolution:a)In a certain primary school 50 pupils were selected to form three schools teams of football, volleyball and basketball as follows30 pupils formed a football team20 pupils formed a volleyball team25 pupils formed a basketball team14 play both volleyball and basketball18 pupils play football and basketball8 pupils play all of the three games7 pupils play football onlyRepresent this information in a Venn diagramSolution:A and B are sets such that n(AB)=4 and n(AB)=6 if A has 4 elementsa) How many elements are there in B?b) Which set is the subset of the otherSolution:(a). 6 elements are in B(c). A⊂BIn general the number of elements in two sets is connected by the formulan(AB)= n(A) + n(B) – n(AB)Exercise:Represent the following in Venn diagramsa) A={a, b, c, d}b) A⊂B c) A= {a, b, c} and B= {a, b, c} d) A= {1, 2, 3} and B= {4, 6, 8}Write in words the relationship between the two sets shown in the figure below-Their relationship is A⊂BDescribe in set notation the meaning of the shaded regions in the following Venn diagramsa) ABA= {a, b, c}b) ABCB= {a, b, c}In a boys school of 200 students, 90 play football, 70 play basketball, and 30 play Tennis. 26 play basketball and football, 20 play basketball and Tennis, 16 play football and Tennis, while 10 play all three games. How many students in school play none of the three games4+10+34+6+10+16+58+N= 200138+N= 200N=200-138N=6262 students play none of the gamesCOMPLEMENT OF A SETIf A is a subset of a universal set, then the compliment of set A may be represented in a Venn diagramExampleShow in a Venn diagram that (AUB)’Solution:(AB)΄ (AB)’ = members of outside ABAB’Represent AÎ,, µ in a Venn diagram and shade the required regionA is a subset of Universal setWORD PROBLEMSExamplesIn a certain school of 120 students, 40 learn English, 60 learn Kiswahili and 30 learn both Kiswahili and English. How many students learna) English onlyb) Neither English nor KiswahiliSolution:Let µ = {students in a school}A= {Students learning English}B= {Students learning Kiswahili}a) n(A) – n(AB) = number of students learning English only40 – 30 = 10Therefore the number of students learning English only is 10b) =n(µ)-[n(A)+ n(B)- n(AB)]= 120-[40+60-30]=120-70=5050 students learn neither English nor Kiswahili AlternativelyBy Venn diagrama) 10 students learn English onlyIn a certain school 50 students eat meat, 60 eat fish and 25 eat both meat and fish. Assuming that every students eat meat or fish, find the total members of students in a schoolSolution:Let µ= {total number of students}A= {students eating fish}B= {students eating meat}n(AB)= n(A)+ n(B)- n(AB)n(AB)=50+60-25n(AB)= 85 studentsThere are 85 students in a schoolAlternatively By Venn diagram85 students were in schoolThere are 24 men at a meeting, 12 are farmers, 18 are soldiers, 8 are both farmers and soldiersa) How many are farmers or soldiersb) How many are neither farmers nor soldiersSolution:NB; both/and means “intersection”By Venn diagramsa)22 men are soldiers or farmersb) 2 are neither farmers nor soldiersIn an examination, 120 candidates offered math, 94 English and 48 offered both math and English. How many candidates offered English but not math assuming that every candidate offered one of the subjects or both math and EnglishSolution:By Venn diagram46 students offered English but not mathEXERCISEA class shows that 15 of the students play basketball, 11 play netball and 6 play both basketball and netball. How many students are there in a class? If every student plays at least one gameSolution:n(AB)= n(A)+ n(B)- n(AB)n(AB)=15+11-6n(AB)=20There are 20 students in the classIn a class of 20 pupils, 12 pupils study English but not History, 4 study History but not English and 1 who study neither English nor History. How many study HistorySolution:Let A= {pupils who study English}B= {pupils who study History}12 + x + 4 = 20X = 3History = x + 4 =77 pupils study HistoryAt a certain meeting 30 people drank Pepsi, 60 drank Coca-Cola, and 25 drank both Pepsi and Coca-Cola. How many people were at the meeting assuming that each person took Pepsi or Coca-ColaSolution:Let A= {drank Pepsi}B= {drank Coca-Cola}n(AB)= n(A)+n(B)-n(AB)=30+60-25=6565 people were at the meetingRepresent (AB) (BC) on a Venn diagramRepresent (AB) CIf set A and B have the same common elements representa) ABb) AB in a Venn diagram In a school of 160 pupils, 50 have bread for breakfast and 80 have sweet potatoes. How many pupils have neither Bread nor potatoes assuming that none take bread and sweet potatoes30 pupils have neither Bread nor PotatoesEvery Man in a certain club owns a Land Rover or a car. 23 men own Land Rover, 14 own cars and 5 owns both Land Rovers and cars. How many men are in a club?32 men were in the clubIn a certain street of 200 houses. 170 have electricity and 145 have glass doors. How many houses have both electricity and glass doors, assuming that each houses has either a glass door or electricity or both170-x + x + 145-x = 200170+145+x-x-x = 200315-x = 200X=315-200X=115115 houses has both electricity and glass doorsREVISION EXERCISEHow many subset are there in A= {a, b, c, d, e, f, g}Solution:Since n(A) = 7 thenFrom 2n=27=128Set A has 128 subsetsList all the subsets of A= {2, 4, 6}Solution:n(A)= 32n=23= 8The subsets are { }, {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}If µ= {a, b, c, d, e}, B= {e, d}, A= {a, b, c} list the elements of B’a) B’= {a, b, c}b) Find A΄B΄Solution:A΄= {d, e}B΄= {a, b, c}A΄B΄ = { }c) A(B΄A΄) = {a, b, c}Draw a Venn diagram and shade the required region of the followinga) A΄Bb) B΄A΄c) ABIn a group of 29 tourists from different countries, 17 went to Manyara national park, 13 to Mikumi national park and 8 went neither Mikumi nor Manyara national park. How many tourists went to both placesTo find x17-x + x + 13-x + 8 = 2938-x = 29X=9:. 9 tourists went both placesFrom the figurea) List the members of set AA= {1, 2, 5, b}b) List the members of set CC= {1, 4, a, b} ALL NOTES FOR ALL SUBJECTS QUICK LINKS:AGRICULTURE O LEVEL PURE MATHEMATICS A LEVELBAM NOTES A LEVELBASIC MATH O LEVELBIOLOGY O/A LEVELBOOK KEEPING O LEVELCHEMISTRY O/A LEVELCIVICS O LEVELCOMPUTER(ICT) O/A LEVELECONOMICS A LEVELENGLISH O/A LEVELCOMMERCE O/A LEVELACCOUNTING A LEVELGENERAL STUDIES NOTESGEOGRAPGY O/A LEVELHISTORY O/A LEVELKISWAHILI O/A LEVELPHYSICS O/A LEVELMOCK EXAMINATION PAPERSNECTA PAST PAPERS Basic Mathematics Study Notes Form 2 Basic Mathematics Study Notes Msomi Maktaba All Notes FORM 2HistoryMATHEMATICSPost navigationPrevious postNext postRelated Posts Basic Mathematics Study Notes Form 4 Mathematics – PROBABILITY November 13, 2018August 17, 2024PROBABILITY Defn: Probability is a branch of mathematics which deals with and shows how to measure these uncertainties of events in every day life. It provides a quantitative occurrences and situations. In other words. It is a measure of chances. Probability set and Event Suppose that an experiment of tossing… Read More Msomi Maktaba All Notes Matokeo Simba vs Singida FG: Who Will Advance? February 3, 202410th Jan 2024: Can Singida FG turn the tide against Simba SC in the semi final of Mapinduzi Cup? Come experience the intense emotions with the rematch. Matokeo Simba vs Singida FG Matokeo Simba vs Singida Fountain Gate: Excitement Builds for Today’s Semi-final The semi-final of the Mapinduzi Cup, featuring… Read More Form 4 Physics Notes PHYSICS FORM FOUR TOPIC 5: ELECTRONIC November 6, 2018February 13, 2019Semi Conductors The Concept of Energy Band in Solids Explain the concept of energy bands in solids In solid-state physics, the electronic band structure (or simply band structure) of a solid describes those ranges of energy that an electron within the solid may have (called energy bands, allowed bands, or… Read More Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment *Name * Email * Website Save my name, email, and website in this browser for the next time I comment. Δ
Basic Mathematics Study Notes Form 4 Mathematics – PROBABILITY November 13, 2018August 17, 2024PROBABILITY Defn: Probability is a branch of mathematics which deals with and shows how to measure these uncertainties of events in every day life. It provides a quantitative occurrences and situations. In other words. It is a measure of chances. Probability set and Event Suppose that an experiment of tossing… Read More
Msomi Maktaba All Notes Matokeo Simba vs Singida FG: Who Will Advance? February 3, 202410th Jan 2024: Can Singida FG turn the tide against Simba SC in the semi final of Mapinduzi Cup? Come experience the intense emotions with the rematch. Matokeo Simba vs Singida FG Matokeo Simba vs Singida Fountain Gate: Excitement Builds for Today’s Semi-final The semi-final of the Mapinduzi Cup, featuring… Read More
Form 4 Physics Notes PHYSICS FORM FOUR TOPIC 5: ELECTRONIC November 6, 2018February 13, 2019Semi Conductors The Concept of Energy Band in Solids Explain the concept of energy bands in solids In solid-state physics, the electronic band structure (or simply band structure) of a solid describes those ranges of energy that an electron within the solid may have (called energy bands, allowed bands, or… Read More