2.+MATHEMATICS+AS+LANGUAGE.pdf

Our lady of Fatima university COLLEGE OF ARTS AND SCIENCES Pampanga Campus Mathematics as Language OBJECTIVES ▪ Mathematics as Language ▪ Sets ▪ Elementary Logic  Mathematics as a LANGUAGE Language,...

Our lady of Fatima university COLLEGE OF ARTS AND SCIENCES Pampanga Campus Mathematics as Language OBJECTIVES ▪ Mathematics as Language ▪ Sets ▪ Elementary Logic  Mathematics as a LANGUAGE 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.  Mathematics as a LANGUAGE Language itself is: Text PRECISE CONCISE POWERFUL It can make very fine It can briefly express long It gives upon expressing distinctions among set of sentences complex thoughts symbols  Mathematics is a SYMBOLIC LANGUAGE Symbols commonly used in Mathematics are:  Mathematics is a SYMBOLIC LANGUAGE ENGLISH Language vs. MATHEMATICS Language: English Mathematics SYMBOLS English Alphabet and English Alphabet, Numerals, punctuation Greek Letters, Grouping Symbols, Special Symbols Name Noun Expressions Complete Thought Sentence Equation Action Verbs Operations and other actions (simplify, rationalize) What is in a sentence Verbs Equality, inequality, membership in a set  Mathematics is a SYMBOLIC LANGUAGE Comparison: English VS Mathematics Language I. Name English Example: a. Carol Mathematics Example: a. 5 b. 1.2 + 6 c. 3x – 3 d. ordered pair (x,y) e. a function f(x)  Mathematics is a SYMBOLIC LANGUAGE Comparison: English VS Mathematics Language II. Complete Thought English Example: a. He loves Mathematics. Mathematics Example: a. x = 5 b. 1 + 3 = 4 c. 3x – 3 = 9 d. 2x ≠ 8  Mathematics is a SYMBOLIC LANGUAGE Comparison: English VS Mathematics Language III. Synonyms English Example: a. Group is to Association Mathematics Example: a. 3 + 5 and 8 b. 1 = 1 c. ½ + ½  Mathematics is a SYMBOLIC LANGUAGE 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  Mathematics is a SYMBOLIC LANGUAGE Translation: English to Math Language and vis a vis 1. English to Math Six less than twice a number is forty-five 2x - 6 = 45 A number minus seven yields ten x – 7 = 10 A total of six and some number 6+x Twelve added to a number x + 12 Eight times a number is forty-eight 8x = 48 The produce of fourteen and a number 14x  Mathematics is a SYMBOLIC LANGUAGE Translation: English to Math Language and vis a vis 1. English to Math Twice a number minus eight 2x – 8 The quotient of a number and seven is two x/7 = 2 Three-fourths of a number ¾x The product of a number and ten is eighty 10x = 80 Eight less than a number is five x–8=5. How many times does five go into 20/5 = x twenty?  Mathematics is a SYMBOLIC LANGUAGE Translation: English to Math Language and vis a vis 2. Mathematics to English Choose a quantity to be represented by a variable, then write the mathematical expression for each number. a. A three-digit numbers whose hundreds digit is half the tens digit, and the tens digit is 2 more than the unit's digit. ½ (x + 2) (x + 2) (x)  Mathematics is a SYMBOLIC LANGUAGE Remember this key words! Addition (+) increased by more than combined, together total of sum, plus added to comparatives ("greater than", etc)  Mathematics is a SYMBOLIC LANGUAGE Remember this key words! Subtraction (-) decreased by minus, less difference between/of less than, fewer than left, left over, after save (old-fashioned term) comparatives ("smaller than", etc)  Mathematics is a SYMBOLIC LANGUAGE Remember this key words! Multiplication (.) (x) () of times, multiplied by product of increased/decreased by a factor of (this last type can involve both addition or subtraction and multiplication!) twice, triple, etc each ("they got three each", etc)  Mathematics is a SYMBOLIC LANGUAGE Remember this key words! Division (a/b) (a:b) (÷) per, a out of ratio of, quotient of percent (divide by 100) equal pieces, split average  Mathematics is a SYMBOLIC LANGUAGE Remember this key words! Equals (=) is, are, was, were, will be gives, yields sold for, cost  Mathematics is a SYMBOLIC LANGUAGE Watch more and Learn More! https://www.you tube.com/watch ?v=v7vBYfvLMDk OBJECTIVES ▪ Mathematics as Language ▪ Sets ▪ Elementary Logic The language of Sets Mathematical Symbol for Sets A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics The language of Sets Mathematical Symbol for Sets The language of Sets Mathematical Symbol for Sets The language of Sets Mathematical Symbol for Sets The language of Sets Mathematical Symbol for Sets  RECITATION Multiple Choice: Basic concepts: Select the best answer to the following multiple-choice questions about basic concepts of sets. OBJECTIVES ▪ Mathematics as Language ▪ Sets ▪ Elementary Logic  Elementary Logic Logic and Symbols Logic serves as a set of rules that govern the structure and presentation of mathematical proofs. It allows us to determine the validity of arguments in and out of mathematics.  Elementary Logic Logic and Symbols A proposition is a statement that is, by itself, either true or false. They can be expressed in symbols P, Q, R, or p, q, r. Types of Propositions: a. Simple – means single idea statement b. Compound – conveys two or more ideas Elementary Logic Logic and Symbols Elementary Logic Logic and Symbols  Elementary Logic Logic and Symbols Logical Connectives STATEMENTS CONNECTIVES SYMBOLIC TYPE OF FORM STATEMENTS Not P Not ~P Negation P and Q And P∧Q Conjunction P or Q Or P∨Q Disjunction If P, the q If…Then P →Q Conditional P if and only if q If and only if P Q Bi-conditional  Elementary Logic Logic and Symbols QUANTIFIER 1. Universal Quantifier “for all” or “for every”, denoted by ∀ 2. Existential Quantifier “there exists”, denoted by ∃ Elementary Logic Logic and Symbols  Elementary Logic Logic and Symbols Let’s consider a propositional language where: p means “Paola is happy”, q means “Paola paints a picture”,  Elementary Logic Logic and Symbols Let: A =“Aldo is Italian” B =“Bob is English”. Formalize the following sentences: 1. “Aldo isn’t Italian” 2. “Aldo is Italian while Bob is English” 3. “If Aldo is Italian then Bob is not English” 4. “Aldo is Italian or if Aldo isn’t Italian then Bob is English” 5. “Either Aldo is Italian and Bob is English, or neither Aldo is Italian nor Bob is English” GO FLEX!