HBMT 4203


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HBMT 4203

  2. 2. INTRODUCTIONSSETS What is sets in mathematics? A set is a collection of distinct objects, considered as anobject in its own right. Sets are one of the most fundamental concepts in mathematics.Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics,and can be used as a foundation from which nearly all of mathematics can be derived. Inmathematics education, elementary topics such as Venn diagrams are taught at a young age,while more advanced concepts are taught as part of a university degree.SETS THEORY Set theory is the branch of mathematics that studies sets, which are collections ofobjects. Although any type of object can be collected into a set, set theory is applied mostoften to objects that are relevant to mathematics. The language of set theory can be used inthe definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor and Richard Dedekindin the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systemswere proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, withthe axiom of choice, are the best-known. Concepts of set theory are integrated throughout the mathematics curriculum in theUnited States. Elementary facts about sets and set membership are often taught in primaryschool, along with Venn diagrams, Euler diagrams, and elementary operations such as setunion and intersection. Slightly more advanced concepts such as cardinality are a standardpart of the undergraduate mathematics curriculum. Set theory is commonly employed as a foundational system for mathematics,particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond itsfoundational role, set theory is a branch of mathematics in its own right, with an activeresearch community. Contemporary research into set theory includes a diverse collection oftopics, ranging from the structure of the real number line to the study of the consistency oflarge cardinals.
  3. 3. SETS HISTORY Mathematical topics typically emerge and evolve through interactions among manyresearchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor:"On a Characteristic Property of All Real Algebraic Numbers".[1][2] Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in theWest and early Indian mathematicians in the East, mathematicians had struggled with theconcept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the19th century. The modern understanding of infinity began in 1867-71, with Cantors work onnumber theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantorsthinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day. While KarlWeierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder ofmathematical constructivism, did not. Cantorian set theory eventually became widespread,due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, hisproof that there are more real numbers than integers, and the "infinity of infinities" ("Cantorsparadise") the power set operation gives rise to. The next wave of excitement in set theory came around 1900, when it was discoveredthat Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes.Bertrand Russell and Ernst Zermelo independently found the simplest and best knownparadox, now called Russells paradox and involving "the set of all sets that are not membersof themselves." This leads to a contradiction, since it must be a member of itself and not amember of itself. In 1899 Cantor had himself posed the question: "what is the cardinalnumber of the set of all sets?" and obtained a related paradox. The momentum of set theory was such that debate on the paradoxes did not lead to itsabandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in thecanonical axiomatic set theory ZFC, which is thought to be free of paradoxes. The work ofanalysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory.Axiomatic set theory has become woven into the very fabric of mathematics as we know ittoday.
  4. 4. SETS DEFINITION Georg Cantor, the founder of set theory, gave the following definition of a set at thebeginning of his Beiträge zur Begründung der transfiniten Mengenlehre.A set is a gathering together into a whole of definite, distinct objects of our perception and ofour thought - which are called elements of the set. The study of algebra and mathematics begins with understanding sets. A set issomething that contains objects. To be contained in a set, an object may be anything that youwant to consider. An object in a set may even be another set that contains its own objects. Ineveryday language, a set can also be called a “collection” or “container”, but in mathematics,the term set is preferred. The objects that a set contains are called its members. The elements or members of a set can be anything: numbers, people, letters of thealphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets Aand B are equal if and only if they have precisely the same elements. As discussed below, the definition given above turned out to be inadequate for formalmathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic settheory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basicproperties are that a set "has" elements, and that two sets are equal (one and the same) if andonly if they have the same elements.CONCEPTSA set A consists of distinct elements :If such elements are characterized via a property E, this is symbolized as follows: satisfies property E}.
  5. 5. The following notations are commonly used: notation meaning is element / member of is not element / member of is a subset of is a strict subset of number of elements in empty setIf ( ), is called a finite (infinite) set.Two sets are called equipotent, if there exists a bijective map between their elements ( for finite sets and ).The set of all subsets of is called power set, i.e. . . Inparticular, we have and . Moreover, .The following sets are standardly denoted by the respective symbols: natural numbers: integers: rational numbers: real numbers: complex numbers:The following notations are also commonly used and asas well as , ,,, ,, respectively.
  6. 6. OPERATIONS ON SETSThe following operations can be applied to sets and : Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} . Set difference, complement of U and A, denoted U A is the set of all members of U that are not members of A. The set difference {1,2,3} {2,3,4} is {1} , while, conversely, the set difference {2,3,4} {1,2,3} is {4} . When A is a subset of U, the set difference U A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U A, particularly if U is a universal set as in the study of Venn diagrams. Symmetric difference of sets A and B is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B) (A ∩ B). Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B. Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } . Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers.
  7. 7. The so- called Venn diagrams illustrate the set operations. union: Intersection: difference: symmetric difference:If , some of the above diagrams are identical to one another: Union: intersection: complement set:
  8. 8. LESSON PLAN:Day : ThursdayDate : 30 June 2011Class : 4 BukhariSubject : Mathematics :Time : 10.00 am – 11.20 amDuration : 80 minutesLearning Area : 3) SetsLearning objectives : Students will be taught to : 3.1 Understand the concept of setsLearning outcomes : Students will be able to : (i) represents sets by using Venn diagramsTeaching aids : Manila cards ((Closed Geometrical shaped), activity sheets, mahjong papers, quiz papers, LCD, worksheets.Attitudes and Values : Patient, self-confident, concentrate, cooperative, follow instructions, honesty, carefulThinking Skills : Categorize, recognize the main idea, making sequence to represent sets by using Venn diagram, locate and collect relevant information, analyze part / whole relationships, reflection.Previous Knowledge : i) sort given objects into groups ii) define sets by description and using set notation iii) identify whether a given object is an element of a set
  9. 9. TEACHING AND LEARNINGSTEPS CONTENTS NOTES ACTIVITIESStep 1 Parts of “Definition of Set” Teacher shows an example Teaching Revision Categorize set A on whiteboard. Aids: Manila Cards Example: Students pay attention on the A = {Factors of 30) example written on the whiteboard. Values: A = {1,2,3,5,6,10,15,30} Self Teacher wants the students to look at confident, the Set A and try to list out all the honest, elements by using set notation. patient. Students try to list out all the Thinking elements of Set a by using set Skills: notation. Categorize Teacher calls a student to give an answer on the whiteboard. A student tries to give an answer. Teacher checks the answer.
  10. 10. STEPS CONTENTS TEACHING AND LEARNING NOTES ACTIVITIES Method:Step 2 Venn Diagram Teacher introduces Venn diagram to Explanation Besides the methods of the students and explains that it is description and set notation, easier to see which group each Teaching sets can be representing by element belongs to in a Venn Aids: using Venn Diagrams. diagram. Manila cards (Closed Closed Geometrical Shapes Students pay attention on Geometrical explanation and the given examples Shapes) of Closed Geometrical Shapes Circle Oval Rectangle Values: Teacher stress that rectangle are Concentrate usually used to represent the set which contain all the elements that Square Triangle Hexagon Thinking are discussed and the circles or Skills: enclosed curves to represent each set Recognize within it. the main idea, Teacher gives examples of Closed making Geometrical Shaped. sequence to represent sets by using Venn diagram
  11. 11. STEPS CONTENTS TEACHING AND LEARNING NOTES ACTIVITIES Step 2 Examples: Then, teacher uses the example from Vocabulary:(continued) induction set and teaches the students Set (1) A={1, 2, 3, 5, 6, 10, 15, 30} the steps to draw a Venn diagram to Element Each ‘dot’ A •1 •2 represents represent set A as listed below: Description •3 •5 •6 one element 1. Draw a circle. Label •10 •15 •30 2. Represent set A by Set notation labeling the circle as A. Denote (2) 3. Determine the number of Venn B={a, b, c} elements in set A, and Diagram B •a represent each of them Empty set •b with a dot inside the circle. Equal sets •c Subset Students concentrate on showing set Universal set (3) Q={Multiples of 3 between in Venn diagram. Complement 8 and 18} of a set Q={9, 12, 15} Teacher reminds students to put a Intersection Q •9 •12 “dot” to present one element and Common •15 label the set. elements Students remember the important point. Teacher shows another two examples - (2) and (3). Teacher asks a student to put elements into diagram.
  12. 12. STEPS CONTENTS TEACHING AND LEARNING NOTES ACTIVITIES Student tries to put elements into diagrams. For example no. (3), teacher asks a student to list out elements first then put the elements into Venn diagram. Another student has to do example no. (3). Teacher gives time to copy notes. Students copy the notes.Step 3 Teaching Progression After copy the given notes, teacher Values: enhance students understanding with Cooperative, Teacher conduct the group conduct group activity calls “fast and Follow activity to further enhance correct”. Instructions student‟s understanding about the lesson learnt today. Teacher explains the rules of the Teaching group activity: Aids: Group Activity “Group that can answer all from 4 Activity 1. Given that W is a set questions fast and correct, will be the sheets, representing days of a week, winner”. Mahjong draw a Venn diagram Paper representing the elements of Students follow the rules in the W. activity.
  13. 13. STEPS CONTENTS TEACHING AND LEARNING NOTES ACTIVITIES 2. Teacher observes students to do the Thinking If, activity. Skills: s x : x is an odd numbered Locate and 20 x 3o , Can you Students ask questions if not collect understand. relevant draw a Venn diagram to information. represent the elements in Teacher wants one representative this set? from each group to come up randomly to check the solution from 3. Draw a Venn diagram to other group on the mahjong paper. represent the set given Example: every first member of the below. group will check number 1, second member from each group to do P x : x is a prime number ? number 2 and so on. i) 1 x 10 ii) 41 x 50 Each group writes down their solution on the mahjong paper, 4. Given that B is the set of depends on the problem solving. common factors of 24 and 36. Finally, teacher discusses the Draw a Venn diagram of set solution with the students. B. Students do the corrections if they make any mistakes.
  14. 14. STEPS CONTENTS TEACHING AND LEARNING NOTES ACTIVITIESStep 4 Quiz Teacher distributes each student a Teaching (1) Given that P = {1, 2, 3, 4, quiz paper (with two questions). Aids: 5}. Represent set P by Quiz Paper, using Venn diagram. Students get a piece of quiz paper. LCD Solution: Teacher asks students try to draw a Values: diagram without asking friends or Honest, P •1 teacher. Careful •2 •3 •4 •5 Students try to draw a Venn diagram Thinking by themselves. Skills: (2) If R = {x : 30 ≤ x ≤ 40, x is Analyze a multiple of 3}, can you (Teacher do not forget gives guide relationships. draw a Venn diagram to line to the students) represent the elements in this set? Teacher observes students to do quiz. Solution: R = {30, 33, 36, 39} Teacher collects papers after three minutes. R •30 •33 •39 Students pass up quiz papers. •36 Teacher discusses the answers for the quiz with the students and guides them. Students respond to the teacher and listen to the answer.
  15. 15. STEPS CONTENTS TEACHING AND LEARNING NOTES ACTIVITIES Step 5 Summary and exercises Teacher asks students to make a Teaching on Venn diagram. summary of the day‟s lesson. Aids:Conclusion Worksheets Students make summary of the day‟s Venn Diagram lesson. Values: Self Teacher reminds students what they Confident Representing in have learnt how to draw a Venn Closed Geometrical Shaped diagram to represent the elements of Thinking a set. Skills: Reflections All the elements Teacher will stress that to put a in a SET “dot” to present one element. Students remember the important Each element will have points. a DOT beside it (left) Teacher distributes the activity sheets. Students do the worksheets and exercises given.
  17. 17. WORKSHEETS 1NAME:_____________________________________ DATE: ______________FORM: 4 _________________Answer all questions.1. Given that Z = {multiple of 3}, determine if the following are elements pf set Z. Fill in thefollowing boxes using the symbol or .a) 52 Z b) 18 Z c) 69 Z2. Y is a set of the months that start with the letter “M”. Define set Y using set notation.3. Determine whether 8 is an element of each of the following set.a) {2,4,6,8}b) {Multiples of 4}c) {LCM of 2 and 4)
  18. 18. d) {HCF of 4 and 8}4. State whether each of the following sets is true or false.a) 4 {Common factors of 8 and 12}b) 1 {Prime number}5. State the number of elements of each of the following.a) X = {Cambodia, Singapore, Malaysia, Indonesia, Thailand}b) Y = {3,6,9, …21)c) Z = {Integers between -3 and 4, both are inclusive}
  19. 19. WORKSHEETS 2NAME:_____________________________________ DATE: ______________FORM: 4 _________________Answer all questions.1. Draw a Venn diagram to represent each of the following:a) F = {1,3,5,7}b) M = {Pictures of durian, mangosteen, mango, starfruit}c) N = {The first 5 prime numbers}2. Use the notation {} to represent set A, B and C.a) A 1 4 9
  20. 20. b) B h j k l m n p rc) C 100 250 150 2003. State n(A) whena) A = {The letters in the word „SCIENCE‟}b) A = {x : 5 x < 30 where x is not an odd number}c) A is a set of perfect square numbers between 0 to 90.
  21. 21. CONCLUSION This assignment shows us the introduction that explain on what is sets, the sets theory