GED03 Chapter 2 (part1) PDF

Chapter II Mathematical Language and Symbols GNED 03 MATHEMATICS IN THE MODERN WORLD Intended Learning Outcome discuss the language, symbols, and AFTER THE STUDENTS HAVE GONE THROUGH conventions of mathematics THIS CHAPTER, THEY SHOULD BE ABLE...

Chapter II Mathematical Language and Symbols GNED 03 MATHEMATICS IN THE MODERN WORLD Intended Learning Outcome discuss the language, symbols, and AFTER THE STUDENTS HAVE GONE THROUGH conventions of mathematics THIS CHAPTER, THEY SHOULD BE ABLE TO: explain the nature of mathematics as a language perform operations on mathematical expressions correctly; and acknowledge that mathematics is a useful language Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World  Language THE SYSTEM OF WORDS, SIGNS AND SYMBOLS WHICH PEOPLE USE TO EXPRESS IDEAS, THOUGHTS AND FEELINGS Mathematical Language and Symbols | GNED03  Mathematical language THE SYSTEM USED TO COMMUNICATE MATHEMATICAL IDEAS INCLUDES A LARGE COMPONENT OF LOGIC, NUMBERS, MEASUREMENT, SHAPES, SPACES, FUNCTIONS, PATTERNS, DATA, AND ARRANGEMENTS. Mathematical Language and Symbols | GNED03 Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World CHARACTERISTICS OF MATHEMATICAL LANGUAGE mathematical language is non-temporal. mathematical language is devoid of emotional content mathematical language is precise Mathematical Language and Symbols | GNED03 Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World MATHEMATICAL EXPRESSIONS AND SENTENCES Math words, expressions and sentences can help students explain what they think. Addition, subtraction, multiplication, and division are operations that can make up a mathematical expression  MATHEMATICAL EXPRESSION MATHEMATICAL SENTENCE Expresses an incomplete thought Expresses a complete thought Contains no relation symbol Contains a relation symbol A mathematical expression is only A mathematical sentence is simplified simplified and solved Mathematical Language and Symbols | GNED03 MATHEMATICAL EXPRESSION MATHEMATICAL SENTENCE  Multivariate mathematical expressions have more than one variable. x THIS SYMBOL IS RARELY USED TO SHOW MULTIPLICATION example 5xy + 9x – 12 31abc y / 3x  MATHEMATICAL EXPRESSIONS TERMS VARIABLE NUMERICAL (LITERAL COEFFICIENT) COEFFICIENT CONSTANT separated from other terms with either plus represents the unknown and The number with the Any single number or minus signs makes use of letters variable  CONSTANT NUMERICAL COEFFICIENT 10x + 11 LITERAL COEFFICIENT MONOMIAL 2a BINOMIAL 5x + 12y TRINOMIAL 3x + 2y – 36 A MATHEMATICAL EXPRESSION POLYNOMIAL WITH MORE THAN TWO TERMS  MATHEMATICAL SENTENCE COMBINES TWO MATHEMATICAL EXPRESSIONS USING A COMPARISON OPERATOR EQUAL, NOT EQUAL, GREATER THAN, GREATER THAN OR EQUAL TO, LESS THAN, AND LESS THAN OR EQUAL TO RELATION SYMBOLS EQUATION INEQUALITY the signs which convey a mathematical a mathematical equality or inequality expression containing the expression containing the equal sign inequality sign EXAMPLES OF EQUATIONS EXAMPLES OF INEQUALITIES: 4x + 3 = 19 15x – 5 < 3y 6y – 5 = 55 18 > 16.5 10 + 1 = m 99 < x (x+y+z) / 2 = 3 1 < x < 10 58 – q = 25 a + b + c ≤ 999  open sentence IT USES VARIABLES, MEANING THAT IT IS NOT KNOWN WHETHER THE MATHEMATICAL SENTENCE IS TRUE OR FALSE closed sentence A MATHEMATICAL SENTENCE THAT IS KNOWN TO BE EITHER TRUE OR FALSE EXAMPLES OF OPEN SENTENCE: ▪ 2xy < 3y ▪ 18w > 16.5 ▪ 3(m + n) = 100 ▪ 8ab – c = 1 ▪ 4 – 3 = v ▪x+y=5 ▪ The obtuse angle is N degrees. ▪ 25m = n ▪ abc = 4 ▪ 3x + 3y – 4z = 11 EXAMPLES OF TRUE CLOSED SENTENCE: ▪ 2(x + y) = 2x + 2y ▪ 18 (2) > 16.5 ▪ 3(m + n) = (m + n) + (m + n) + (m + n) ▪ 8c – c = 7c ▪ 9 is an odd number. ▪ √25 = 5 ▪ 10 – 1 = 9 ▪6–6=0 ▪ The square root of 4 is 2. ▪ g + g + 100 = 2g + 100 EXAMPLES OF FALSE CLOSED SENTENCE: ▪ 9 is an even number. ▪4+4=5 ▪ 10 – 1 = 8 ▪6–6=–1 ▪ The square root of 4 is 1. ▪ d + 2d = 2d2 ▪ y 0 = 2 ▪ (xyz)2 = 2xyz ▪ x + 2x + 3x = 10x Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World  CONVENTIONS IN THE MATHEMATICAL LANGUAGE CONTEXT CONVENTION refers to the particular topics being studied and a technique used by mathematicians, engineers, it is important to understand the context to scientists in which each particular symbol has understand mathematical symbols. particular meaning  GREEK AND LATIN LETTERS ARE USED AS SYMBOLS FOR PHYSICAL QUANTITIES AND SPECIAL FUNCTIONS; AND FOR VARIABLES REPRESENTING CERTAIN QUANTITIES THE POSITION OF NUMBERS AND SYMBOLS IN RELATION TO EACH OTHER HAS A BEARING ON THEIR MEANINGS. USAGE OF SUBSCRIPTS AND SUPERSCRIPTS IS ALSO AN IMPORTANT CONVENTION. GUIDELINES IN TRANSLATING STATEMENTS TO MATHEMATICAL EXPRESSION/SENTENCE READ AND UNDERSTAND THE PROBLEM/TEXT ENTIRELY SET VARIABLES FOR THE UNKNOWN VALUES LOOK FOR “KEYWORDS” ASSOCIATED WITH MATHEMATICAL OPERATIONS FAMILIARIZE YOURSELF WITH COMMONLY USED MATHEMATICAL OPERATORS AND SYMBOLS Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World VARIABLE sometimes thought of as a mathematical "John Doe" because you can use it as a placeholder when you want to talk about something but either you imagine that has one or more values but you don't know what they are.  EXAMPLE Is there a number with following property: doubling it and adding 3 gives the same result as squaring it?  EXAMPLE Is there a number with following property: doubling it and adding 3 gives the same result as squaring it? Is there a number x with property that 2x + 3 = x²?  EXAMPLE No matter what number might be chosen, if it is greater than 2, then its square is greater than 4.  EXAMPLE No matter what number might be chosen, if it is greater than 2, then its square is greater than 4. No matter what number n might be chosen, if n is greater than 2, n² is greater than 4 Exercise USE VARIABLES TO REWRITE THE FOLLOWING SENTENCES MORE FORMALLY. ARE THERE NUMBERS WITH THE ARE THERE NUMBER A PROPERTY THAT THE SUM OF THEIR AND B SUCH THAT A² SQUARE EQUALS THE SQUARE OF + B² = (A+B)²? THEIR SUM? GIVEN ANY REAL NUMBER, ITS FO R ANY REAL SQUARE IS NONNEGATIVE NUMBER R, R²≥0  Set A WELL-DEFINED COLLECTION OF DISTINCT OBJECTS The set of vowels in the English Alphabet The set of counting numbers less than 20 The set of all letters in the word “Philippines”  ELEMENTS Objects or components that make up a set Examples: Let A be a set containing all the vowels in the English Alphabet. With this, the following statements must be true: Letter “u” is an element of set A since letter “u” is a vowel. In symbols, we can express this as u∈A. The same applies to the letters “a”, “e”, “i”, and “o” since all of these are vowels in the English Alphabet. Letter “d” is not an element of set A since letter “d” is a consonant. In symbols, we can express this as d ∉ A. "This applies, as well, to other consonants in the English Alphabet.  CARDINALITY OF A SET refers to the number of elements a set has Examples: Consider the following sets: Set G contains all the “ber” months Set H contains all distinct letters in the word “MATHEMATICS” Set I contains all perfect square numbers less than 100 THERE ARE TWO WAYS TO DESCRIBE A SET, NAMELY: Roster/Tabular Method the elements in the given set are listed or enumerated, separated by a comma, inside a pair of braces. Examples: Let B be a set containing all the prime numbers between 1 and 10 B= {2, 3, 5, 7} THERE ARE TWO WAYS TO DESCRIBE A SET, NAMELY: Rule/Descriptive Method the common characteristics of the elements are defined. This method uses set builder notation where x is used to represent any element of the given set Examples: Let B be a set containing all the prime numbers between 1 and 10 B= {x|x is a prime number between 1 and 10} TWO METHODS OF DESCRIBING SETS Try to answer the following: Let C be a set containing all months with 31 days. Express this set using roster method Let D be a set containing the elements red, yellow, and blue. Express this set using rule method Let E be a set containing the elements 2, 4, 6, and 8. Express this set using rule method Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World  FINITE SETS sets with a limited number of elements; the last element is specified Examples: The set of counting numbers less than 5 The set of all letters in the English alphabet V= {a, e, i, o, u}  INFINITE SETS sets with an unlimited/infinite number of elements; the last element cannot be specified Examples: The set of the stars in the sky The set of all real numbers E = {2, 4, 6, 8, 10, …} EMPTY SET/NULL SET/VOID SET set that has no elements. It is expressed using the symbols { } or ∅  UNIVERSAL SET the set of all possible elements at a given situation Examples: Let a set be created containing all the possible outcomes generated from rolling a die. With this, we can say that: U= {1, 2, 3, 4, 5, 6} Let a set containing all the possible colors a student can pick from a standard deck of cards be created. With this, we can express the set as: U= {red, black} SET RELATIONS  SUBSET for any arbitrary sets A and B, set A is a subset of B if every element of A is an element of B SUPERSET we can also say that B is a superset of A since B contains every element of A.  IMPROPER SUBSET a subset containing all the elements of a given set A is an improper subset of B A ⊆ B PROPER SUBSET a subset containing elements of a given set A is an proper subset of B A ⊂ B Examples: Consider the following sets: V= {a, e, i, o, u}, W= {a, e, i}, & X= {x|x is a vowel in the English alphabet}. W ⊂V W ⊂X V ⊆X X ⊆V  EQUAL SETS are sets containing the same elements EQUIVALENT SETS are sets with the same number of elements Examples: Consider the following sets: A= {x|x is an odd number between 1 and 9} B={a, b, c} C={3, 5, 7}  JOINT SETS sets with at least one common element. DISJOINT SETS sets with no common elements Examples: Consider the following sets: A={1, 3, 5} B={2, 4, 6} C={2, 3, 4}  UNION Union of Sets A and B (denoted by A ∪ B) is the union of two sets A and B is the set of all elements belonging to either set A or set B In symbol: A ∪ B = {x | x ∈A or x ∈ B} INTERSECTION Intersection of sets A and B (denoted by A ∩ B) is the intersection of two sets A and B is the set of all elements belonging to both set A and set B. In symbol: A ∩ ∈ B = {x | x A and x ∈B} EXAMPLES Consider the following sets: P={1, 3, 5} Q={2, 4, 7} Evaluate the following set operations: R={3, 5, 7} ∪ 1. P Q 2. P∪R 3. P∩R 4. Q∩R 5. P∪Q∪R 6. P∩Q∩R  DIFFERENCE Difference of sets A and B (denoted by A – B) is a set whose elements are found in Set A but not in Set B. In symbol: {x | x ∈ A and x ∉ B} COMPLEMENT Complement of SetA (denoted by A’) is a set whose elements are found in the universal set but not in Set A. In symbol: A’ = {x | x ∈U and x ∉ A} EXAMPLES Consider the following sets: U={0, 1, 2,3 , 4, 5, 6, 7, 8, 9} M={1, 3, 5, 7, 9} Evaluate the following set operations: N={0, 6, 7, 8, 9} 1. M′ 2. N′ 3. M-N 4. N-M ∪ 5. (M N)′ 6. (M∩N)′ Venn-Euler Diagrams (Venn Diagrams) A-B B-A ∪∪ A B C A′ B′ ∩∩ A B C  USING VENN DIAGRAM IN SOLVING PROBLEMS INVOLVING SETS 1. Seventy-five (75) students were asked about their preference over Math and English subjects. It was found out that 40 of them prefers Math, 50 prefers English, and 28 prefers both subjects. Determine the number of students who: a. prefer Math only b. prefer English only c. prefer neither Math nor English subject In order to solve the given problem using Venn Diagram, consider the following steps: 1. Construct a Venn diagram 2. Fill in the Venn Diagram with appropriate values 3. Solve the problem 1. Construct a Venn diagram How many sets are there in the problem? Are the sets given joint or disjoint? 1. Construct a Venn diagram How many sets are there in the problem? Are the sets given joint or disjoint? 2. Fill in the Venn Diagram with appropriate values What is the cardinality of the universe (U)? What are the given values which can provide information about the cardinality of each set? What values should be placed in each region in the Venn Diagram? 1. Construct a Venn diagram How many sets are there in the problem? Are the sets given joint or disjoint? 2. Fill in the Venn Diagram with appropriate values What is the cardinality of the universe (U)? What are the given values which can provide information about the cardinality of each set? What values should be placed in each region in the Venn Diagram? 3. Solve the problem How many students prefer Math only? How many students prefer English only? How many students prefer neither Math nor English? Mathematical Language and Symbols CHAPTER II GNED 03 | Mathematics in the Modern World