Mathematics in Modern World, Lecture notes of Mathematics

Download Mathematics in Modern World and more Lecture notes Mathematics in PDF only on Docsity! 14 For thousand years, mathematicians had developed spoken and written natural languages that are highly effective for expressing mathematical language. This mathematical language has developed and provides a highly efficient and powerful tool for mathematical expression, exploration, reconstruction after exploration, and communication. Its power comes from simultaneously being precise and yet concise. But the mathematical language is being used poorly because of poor understanding of the language. The mathematical language and logical reasoning using that language form the everyday working experience of mathematics. Chapter II MATHEMATICAL LANGUAGE AND SYMBOLS TOPICS 1. Mathematical Language 2. The Language of Sets 3. The Language of Relations and Functions LEARNING OUTCOMES At the end of the lesson, you should be able to:  Discuss the language, symbols, and conventions of Mathematics.  Explain the nature of Mathematics as a language.  Compare and contrast expression and sentences.  Discuss the concept of sets.  Represent sets using roster method and rule method.  Differentiate roster method and rule method.  Differentiate finite set and infinite set.  Discuss different set terminologies.  Solve problems involving sets.  Discuss the concept of relations and functions.  Identify the domain and range of relations.  Identify relations which are functions and not functions.  Represent relations and functions using mapping diagrams. 15 CHARACTERISTICS OF MATHEMATICAL LANGUAGE Notably, mathematics has its own language, much of which we are already familiar with e.g. the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Whether we refer to 0 as "zero," or "nothing" we understand its meaning. There are many symbols in mathematics and most are used as a precise form of shorthand. It is quite important that we familiarize ourselves using these symbols and we understand their meaning. Mathematical language can easily be understood by context and convention. Context is what we are working or the particular topics being studied, while convention is where mathematicians and scientists have decided that particular symbols will have particular meaning. The mathematical language is the system used to communicate mathematical ideas. This language consists of some natural language using technical terms (mathematical terms) and grammatical conventions that are uncommon to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas. The mathematical notation used for formulas has its own grammar and shared by mathematicians anywhere in the globe. The characteristic of mathematical language is being precise, concise, and powerful. 18 The Language of Sets Forget everything you know about numbers. In fact, forget you even know what a number is. This is where Mathematics starts. Instead of math with numbers, we will now think about math with “things”. Set theory is the branch of Mathematics that studies sets or the mathematical science of the infinite. The study of sets has become a fundamental theory in Mathematics in 1870s which was introduced by Georg Cantor (1845-1918), a German mathematician. What is a set? Well, simply put, it’s a collection. A set is a well-defined collection of objects; the objects are called the elements or members of the set. The symbol is used to denote that an object is an element of a set, and the symbol denotes that an object is not an element of a set. SOME EXAMPLES OF SETS:  A = {x ꟾ x is a positive integer less than 10}  B = {-1,-2, -3, -4, -5, -6, -7}  C = {x ꟾ 10 < x < 20}  D = The set of letters in the word dirt.  E = {x ꟾ x is a set of consonant letter in the Englih alphabet} 19 Set can be represented in any one of the following ways or forms. One way is to give a verbal description of its elements. This is known as the Descriptive form of specification. Another is Roster method. This is when the elements of the set are enumerated and separated by a comma. It is also called Tabulation method. Lastly, is the Rule method which describes the elements or members of the set. It is also called Set builder notation. In the rule method or set builder notation ‘ꟾ ’ is read as “such that”. Example 1 A= {x ꟾ 10 < x < 20} We read it as “A is the set of all x such that x is a greater than 10 but less than 20” Example 2 B= { x ꟾ x is a vowel in English alphabet} “A is the set of all x such that x is a vowel in the English Alphabet” 20 Direction: Given below are some sets. Your task is to identify whether the given set is written in descriptive form, in roster method or in rule method. 1. A = {x ꟾ x is a positive integer less than 15} 2. B = The set of odd numbers more than 15 but less than 35. 3. C = {5, 10, 15, 20, 25, 30, 35, 40, 45,..} 4. D = {x ꟾ x is a set of consonant letter in an English alphabet} 5. E = {x ꟾ x is a positive integer greater than 45} 6. F = {m, a, t, h} 7. G = {3,6,9,12,15} 8. H = The set of counting numbers less than 20. 9. I = {x ꟾ x is a whole number greater than 12} 10. J = {x ꟾ 40 > x > 20} Direction: Given below are some sets written in descriptive method. Your task is to write each given sets into its roster method and rule method. 1. Q = The set of all prime numbers less than 100. 2. R = The set of even natural numbers less than 11. 3. S = The set of letters in the word universe. Direction: Given below are some sets written in roster method. Your task is to write each given sets into its descriptive and rule method. 4. K = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} 5. L = {1, 2, 4, 5, 10, 20, 25, 50, 100} 6. O = {e, g, o} Direction: Given below are some sets written in rule method. Your task is to write each given sets into its descriptive method and roster method. 7. P = {x ꟾ 13 < x < 24} 8. Q = {x ꟾ 43 > x > 29} 9. R = {x ꟾ x is a negative integer greater than -14} 10. S = {x ꟾ x is a whole number less than 16} TRY IT YOURSELF! TRY HARDER! 23 Example: Power set is the collection of all subsets of a given set. It is denoted by. Example: SET POWER SET X= {e, f} X = {{e}, {f}, {e, f}, {}} Y= {1, 2, 3, 4} Y = {{1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {2, 3, 4}, {3, 4, 1}, {4, 1, 2}, {1, 2, 3, 4}, {}}. The union of A and B, denoted by A B, is the set of all elements x in U such that x is in A or x is in B. Example: Elements with the same identity in Set A and B is can be written once in AUB. So, we need not to write three 3’s in AUB. The intersection of A and B, denoted by A B, is the set of all elements x in U such that x is in A and x is in B. Example: Two sets are called disjoint (or non-intersecting) if and only if, they have no elements in common. In symbol, A B = . Example: 24 The Cartesian product of sets A and B, written as A x B. Given sets A and B, the Cartesian product of A and B, denoted A x B and read “A cross B” is the set of all ordered pairs (a,b), where a is in A and b is in B. Example: Direction: Given below are some sets. Your task is to identify whether the given set is a finite set or an infinite set. 1. K = {x ꟾ x is a positive integer less than 98} 2. J = {50, 100, 150, 200, 250, 300, 350, 400, 450...} 3. Z = {x ꟾ x is a negative integer less than 0} 4. Y = {l, o, v, e} 5. X = {x ꟾ 104 < x < 150} 6. W= {x ꟾ x is a whole number greater than 1200} 7. V = {-9, -11, -13, -15, -17, -19, -21, -23, -25} 8. U= {x ꟾ 27 > x > 11} 9. T = {x ꟾ x is a set of letters in the alphabet} 10. S = {x ꟾ x > 58} Direction: Given below are some sets. Your task is to identify whether the given set is a unit set or a null set. 1. Y = {16} 2. T = {x ꟾ x is a negative integer greater than 17} 3. R = {x ꟾ x is a whole number greater than 5 but less than 6} 4. A = {x ꟾ x 91 < x < 93} 5. P = {x ꟾ x is a positive integer greater than 10 but less than 11} TRY IT YOURSELF! TRY HARDER! 25 Direction: Given below are some sets. Your task is to identify the cardinality of each set. 6. T = {x ꟾ 34 < x < 45} n(T)= _________ 7. W = {x ꟾ 43 > x > 39} n(W)= _________ 8. O = {x ꟾ x is a negative integer greater than -16} n(O)= _________ 9. S = {x ꟾ x is a whole number less than 19} n(S)= _________ 10. N = {x ꟾ x is a positive integer less than 22} n(N)= _________ Direction: Given below are Set A, Set B, Set C, Set D and Set E. Solve for the following using these sets. A = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60} B = {5, 10, 20, 40, 80, 160} C = {1, 2, 4, 5, 10, 20, 25, 50, 100} D = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} E = {-1, -2, -3, -4, -5, -6, -7, -8, -9, -10} 1. A B = ____________________ 2. A C = ____________________ 3. A D = ____________________ 4. A E = ____________________ 5. B C = ____________________ 6. B D = ____________________ 7. B E = ____________________ 8. C D = ____________________ 9. C E = ____________________ 10. D E = ____________________ ASSESSMENT TIME!