MATHEMATICAL SYSTEM February 24, 2021 postadmin Post in Uncategorized Page 1 of 72 Course Title:MATHEMATICS IN THE MODERN WORLD MODULE 3MATHEMATICAL SYSTEMModule IntroductionHistorically, a mathematical system consists of undefined terms, definitions, postulates oraxioms, propositions or theorems as contained on the classic Book I of the Elements written by aGreek mathematician, Euclid of Alexandria. This treatise systematizes Geometry containing 23definitions, 5 postulates, 5 common notions, and 48 propositions (Baltazar, Ragasa, &Evangelista, 2018).Notably, these geometric concepts were alreadyintroduced and applied by the Egyptians andBabylonians who came before Euclid. The Elementsof Euclid philosophizes these concepts putting theminto more coherent, abstract and consistent ideas. Itestablishes “geometry as a primary inquiry” (YoungOcean, 2019).Nothing has changed about the system that Euclidhas in his geometry up to the present day. All of hispropositions can be reproduced by a contemporarystudent with accuracy equivocal to how Eucliddemonstrated it. Geometry has become a concretemathematical system where the elements of a set areconcreted or illustrated into concrete figures, and onhow these objects of the set are related.However, in this Module, the learners will learn that a mathematical system is also an abstract oralgebraic system with the concept of set as building block. The emphasis on this section is thestudy of the important properties of a mathematical system of a set of integers, the remainderfunction particularly modular arithmetic. LESSON NO.1LESSON TITLEBINARY OPERATIONSSpecific LearningOutcomes:During the learning engagements, the learners should be able to:1. Discuss the nature and properties of a mathematical system throughsome forms of semantic maps;2. Determine the binary properties satisfied of a given mathematicalsystem; and3. Explain your stand on the adage: “Mathematics, as building block oftechnology, revolutionizes society.”TEACHING LEARNING ACTIVITIESPROCESSING1. Defining a Mathematical System. Investigate which among the three ordinary operationsin real number is/are binary – addition, multiplication, division. The following are theinstructions: Perform the following indicated operations on a set of real number , if it exists.a. 1 + 5 (defined, solution exists) f. 6 ÷3(defined, solution exists)b. 7 · 0 (defined, solution exists) h. 0 ÷ 4 (defined, solution exists)c. 3 · 6 (defined, solution exists) i. 0 · 0 (defined, solution exists)d. 0 + 8 (defined, solution exists) j. 4 ÷ 0 (undefined, solution does not exist)e. 5 · 8 (defined, solution exists) h. 0 ÷ 0 (indeterminate, solution does not exist) Euclid of Alexandria and his Book Elements. Photo retrievedfrom https://miro.medium.com/max/500/1*4ijBUB_OSisa7Y2Fmrihw.jpegPage 2 of 72 Considering the entire set , Only Addition and Multiplication are defined to all elements of? (Hence, BINARY OPERATIONS)DEFINITION:A Mathematical System is a set with one or more binary operations defined on it, anda binary operation is a rule that assigns to 2 elements of a set having a unique third element.PROPERTIES OF BINARY OPERATIONS/BINARY PROPERTIES CLOSURE OF A SET (Special Property of Binary Operation)“A set is said to be closed under binary operation * if for any two members from the set, theresult of the binary operation returns a member of the set.”Example: for all elements under the given SET OF INTEGERS (Z),If 5+2=7, then this equation is closed.If 5*2=10, then this equation is also closed.(1) Note that 5 and 2 are members of the set of all integer numbers(2) Performing any binary operation to 5 and 2 will produce 7 (ifaddition) and 10 (if multiplication)… and 7&10 are both integers as well(3) Going back to the definition of CLOSURE, (5+2=7) and (5*2=10)are closed because the 2 members of the integer set (5 and 2) producedanother members of the given set of integers (7 and 10)Determine if the operations and equations below are closed or not:There’s a given set X.X=1, 2, 3, 4, …100Is (25 and 75) closed under the binary operation?Check!By addition: 25+75=100 (25,75, and the third element 100are all members of the set X.Hence, CLOSED under addition)By Multiplication: 25*75= 1875 (25 & 75, are all members of theset X. However, 1875 doesnot belong to set X. Hence,NOT CLOSED undermultiplication)For the remaining 4 basic binary properties…Suppose we consider a finite set S = 1, 2, 3, 4 and a binary operation , and let theoperation be defined on the following table.1 2 3 41 2 1 4 32 1 2 3 43 4 3 2 14 3 4 1 2We could see that the mapping of 1 onto 3 is 4, or simply written as 13 = 4. Similarly, 23 =3. We note that, the operation is closed in S because said operation is defined in all elementsof S. Page 3 of 72 COMMUTATIVITY“A Binary operation * defined on a set R of real numbers is COMMUTATIVE ifa*b=b*a, for all a, b are elements of R”We can also verify that13 = 4 and31 = 4.Also, 24 = 4 and42 = 4.This shows that the operation is commutative…We can verify this furthermore because there is a symmetry that exists with respect to thediagonal.1 2 3 41 2 1 4 32 1 2 3 43 4 3 2 14 3 4 1 2 ASSOCIATIVITY“A Closed Binary operation * defined on a set R of real numbers is ASSOCIATIVE ifa*(b*c)=(a*b)*c, for all a, b, c are elements of R”We also verify whether 1(23) = (12)3 or not.We take the innermost operation first, as it was introduced to you in elementary and highschool.1(23) = (12)313 = 1 34 = 4.Another…3(14) = (31)433 = 442=2 EXISTENCE OF IDENTITY“An identity element of a binary operation is a single element that will return to its originalvalue when the operation is performed.”The identity element of the binary operation * on a set S is wfor which a * w = a, for all a, w are elements of S. Page 4 of 72 Notice that,1 2 = 1 3 2 = 32 2 = 2 4 2 = 4.Thus, 2 is the identity element of the binary operation . EXISTENCE OF INVERSE“The inverse of the binary operation * on a set S is efor which a * e = w, where w is the identity element of * on S,and where all a, e, w are elements of S”We can also decipher from the given operation that: (note that we have previously seen that 2is the identity element of the binary operation on set S)1 1 = 2 3 3 = 22 2 = 2 4 4 = 2.We can say that the inverse of 1, written as 1-1, is 1; the inverse of 2 is 2 or 2-1 = 2. Similarly,3-1 = 3 and 4-1 = 4.FORMATION ACTIVITYAnswer the attached Activity on the Anticipation Guide for Binary Operation, titled as BinaryOperations – “Do Now” retrieved from https://www.easyteacherworksheets.com/math/algebrabinaryoperations.html.2. Based from their answers on the Activity, ask the following:2.1. How important is the concept of set in understanding amathematical system?2.2. Why are the properties of binary operation important inunderstanding the nature of a given mathematical system?2.3. A certain mathematical system may be designed in order tosolve certain issue of humanity. What is your stand on theadage: Mathematics, as building block of technology,revolutionizes society? Explain further your answer.SYNTHESIS ACTIVITY1. Design any semantic map or diagram that would show the relationship of the followingterms. Give a brief description of your map.binary operation set mathematical system closure propertyCommutativity associativity identity inverse2. Complete the given KWL Chart to show what you know, what you wonder and what youhave learned from this lesson.What I KNOW about this lesson? What I WONDER about this lesson? What do I LEARNED from this lesson? Page 5 of 72 ASSESSMENT1. Suppose we consider a finite set X = N, I, K, O and abinary operation ®, and let the operation ® be defined on thefollowing table:®N I K ON K N I OI N K O IK I O K NO O I N KDetermine if the binary operation ® exists and is defined in thegiven set X by verifying all the Binary Properties: CLOSURE COMMUTATIVITY ASSOCIATIVITY EXISTENCE of IDENTITY EXISTENCE of INVERSE2. Why the ordinary division (÷) is not a binary operation in theset of real number?3. Let the operation © be defined on a set of integer Z,such thata © b = 2a – b, for all a, b ZFind the value of3 © (-1) = ________Hint: substitution of values for a and b… Page 6 of 72 4. In relation to number 3, is operation © closed or not? Pleasejustify5. 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 25.1. Is closed? Justify.5.2. Show that is commutative5.3. Does Associativity in exist? Justify.5.4. What is the identity element of the operation ?5.5. What is/are the inverse element(s) of the operation?REFLECTIVE LEARNING6. Cite a certain mathematical system applied to the program orcourse that you are taking. Describe how mathematical systemswork and benefit the humanity. Page 7 of 72 RESOURCES:Baltazar, Ragasa, & Evangelista, Justina(2018). Mathematics inthe Modern World. C and E Publishing Inc.https://study.com/academy/practice/quiz-worksheet-binaryoperation-binary-structure.html.https://www.youtube.com/watch?v=A5bMGt8inM0&list=PL9bjpkdabbOsz314XDGNiS2s7o3ZDQPUJ&index=5https://www.youtube.com/watch?v=jKoMerdR2Ig.https://www.easyteacherworksheets.com/math/algebrabinaryoperations.html.Young Ocean (2019). Math and Magic: Euclid Defines Space.Retrieved on 5 July 2020 fromhttps://medium.com/@sunfaceman/math-and-magiceuclid-defines-space-ea987f61709c.