Unit 2 Nature of Mathematics PDF

Our lady of Fatima university COLLEGE OF ARTS AND SCIENCES Math and Physics Department Pampanga Campus MATM111 Unit 2 Nature of Mathematics: Mathematics Language and Symbols Ronell M. Mangilit, Ph. D. Course Instructor II. Nature of Mathematics...

Our lady of Fatima university COLLEGE OF ARTS AND SCIENCES Math and Physics Department Pampanga Campus MATM111 Unit 2 Nature of Mathematics: Mathematics Language and Symbols Ronell M. Mangilit, Ph. D. Course Instructor II. Nature of Mathematics: Mathematics Language and Symbols uag 2.1 2.2 2.3 Language of Sets Elementary Mathematics Logic Learning Outline  II. Nature of Mathematics: Mathematics Language and Symbols Lesson 2.1: Language of Mathematics At the end of the lesson, the students are expected to: be familiar with the key words used in Mathematics translation; and translate English phrases/sentences into Mathematics expressions/equations. Learning Outcomes Language, a system of conventional spoken, manual (signed), or written symbols by means of which human beings, as members of a social group and participants in its culture, express themselves. Language itself is: PRECISE CONCISE POWERFUL It can make very fine It can briefly express long Because of the relative ease it distinctions among a sentences. gives upon expressing set of symbols. complex thoughts. Commonly Used Symbols in Math Mathematics is a symbolic language. ENGLISH Language vs. MATHEMATICS Language English Mathematics Symbols English Alphabet and punctuation English Alphabet, Numerals, Greek Letters, Grouping Symbols, Special Symbols Name Noun Expression Complete Thought Sentence Equation Action Verbs Operations and other actions (e.g., simplify, rationalize) What is in a sentence Verbs Equality, Inequality, membership in a set Attribute of a Fact or Fiction True or False sentence Synonyms different words but the same meaning the same object but different names Comparison: English VS Mathematics Language I. Name Examples: 1. Carol English 2. loves Mathematics Mathematics Examples: 1. 5 2. 1.2 + 6 3. 3x – 3 4. ordered pair, (x,y) 5. a function, f(x) Comparison: English VS Mathematics Language II. Complete Thought English Example: 1. Carol loves Mathematics. Mathematics Examples: 1. x = 5 2. 1 + 3 = 4 3. 3x – 3 = 9 4. 8 ≠ 7 5. x ≥ 8 Comparison: English VS Mathematics Language III. Synonyms English Examples: 1. Group – Association Mathematics Examples: 1. 1+ 2 + 5 and 8 1 1 4 2. + , and 2 − 1 2 2 4 Some Difficulties in the Math Language 1. Different meaning or use of words in Math and English. “and” is equivalent to plus “is” may have different meaning 2. The different uses of numbers: cardinal, ordinal, or nominal. “cardinal numbers” ones used for counting “ordinal numbers” ones used for telling positions “nominal numbers” used only as a name to identify something not as an actual value or position 3. Mathematical objects may be expressed in many ways such as sets and functions.  KEY WORDS Equals (=) is, are, was, were, will be gives, yields sold for, costs KEY WORDS  TRANSLATION: ENGLISH LAGUAGE TO MATHEMATICAL LANGUAGE Translate each English phrase/sentence into Let x be the unknown number. Mathematical expression/equation. ENGLISH LANGUAGE MATHEMATICAL LANGUAGE 1. Twice a number minus eight 2. The quotient of a number and seven is two. 3. Three-fourths of a number 4. The product of a number and ten is eighty. 5. Eight less than a number is five. 6. Five goes into a number four times 7. Thrice a number subtracted from 10 8. The sum of two consecutive numbers is 15. 9. The square of the sum of 6 and a number II. Nature of Mathematics: Mathematics Language and Symbols uag 2.1 2.2 2.3 Language of Sets Elementary Mathematics Logic Learning Outline  II. Nature of Mathematics: Mathematics Language and Symbols Lesson 2.2: Sets At the end of the lesson, the students are expected to: be familiar with a set and its related concepts; and perform operations of sets – union, intersection, and complement. Learning Outcomes  a. b. c. a. b. c. a. 1. 2. b. c. II. Nature of Mathematics: Mathematical Language and Symbols 2.1 2.2 2.3 Language of Sets Elementary Mathematics Logic Learning Outline Logic ✓ It is the study of reasoning and argument. ✓ It allows us to determine the validity of arguments in and out of mathematics. A proposition is a declarative statement that may express an idea which can be true or false, but not both. They can be expressed in symbols P, Q, R, or p, q, r. Type equation here. a. Simple – means single idea statement Examples: 1. Washington, D.C., s the capital of the United states of America. 2. Eighteen is a perfect square. 3. Nine is a prime number. 4. xyz = yzx b. Compound – conveys two or more ideas Logical Connectives STATEMENT CONNECTIVE SYMBOLIC FORM TYPE OF STATEMENT not P not ∼P Negation P and Q and P∧Q Conjunction P or Q or P∨Q Disjunction If P, then q. If…then P→Q Conditional P if and only if q if and only if P Q Bi-conditional QUANTIFIERS 1. Universal Quantifier - “for all” or “for every”, denoted by ∀ 2. Existential Quantifier - “there exists”, denoted by ∃