PDF - Mathematical Language and Symbols

TOPIC 2 MATHEMATICAL LANGUAGE AND SYMBOLS LEARNING OBJECTIVES: At the end of the lesson, the student will be able to: a. Use Mathematical language appropriately in writing mathematical ideas; b. Represent sentences using set notations; and c. Construct a truth tabl...

TOPIC 2 MATHEMATICAL LANGUAGE AND SYMBOLS LEARNING OBJECTIVES: At the end of the lesson, the student will be able to: a. Use Mathematical language appropriately in writing mathematical ideas; b. Represent sentences using set notations; and c. Construct a truth table for a given compound statement. INTRODUCTION In writing and speaking language of Mathematics, it is important that you know some basic mathematical terms because you will most likely encounter mathematical sentences with complicated structures. One very important element for a student to succeed in mathematics is the ability to communicate effectively in Mathematics (Schiro,1997). Students should be given opportunities to understand the math language in the classroom so that they can correctly read and write mathematical symbols which they can use effectively to solve math problems. Most students have problems understanding mathematical concepts not because they are difficult but because the ideas are presented in the language of math. Thus, familiarization with mathematical language and symbols is important. Time allotment/ duration: 3 hours Core-Related values and Biblical Reflection: Excellence: Accuracy Hebrews 2:14: “Since therefore the children share in flesh and blood, he himself likewise partook of the same things, that through death he might destroy the one who has the power of death, that is, the devil. (ESV) LEARNING CONTENT Topic Content: A. Mathematics as a Language B. Sets and Functions C. Mathematical Logic A. Mathematics as a Language http://www.onemathematicalcat.org/pdf_files/LANG1.pdf The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is: a. precise (able to make very fine distinctions); b. concise (able to say things briefly); and c. powerful (able to express complex thoughts with relative ease) Every language has its vocabulary (the words) and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception. The following are the parts of speech for Mathematics: 1. Numbers- use to represent quantity (these are nouns/objects in the English Language).  2. Operation Symbols- (+, -, ÷, ^ and ˅) can act as connectives in a Mathematical sentence. 3. Relation Symbols- (=, ≥, ≤ and ~) are used for comparison and act as verbs in the mathematical language. 4. Grouping Symbols- (), {} and [], are used to associate groups of numbers and operators. 5. Variables- letters that represent quantities and act as pronouns. English Language to Mathematical Language: Noun to mathematical expressions Examples: 𝑥 + 5; 𝑠𝑖𝑛𝑥; 𝑙𝑜𝑔𝑥 Sentence to mathematical sentence Example: 2𝑥 + 𝑦 = 6 B. Sets and Functions https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch1.pdf A set is a collection of objects, called the elements or members of the set. The objects could be anything (planets, squirrels, characters in Shakespeare’s plays, or other sets) but for us they will be mathematical objects such as numbers, or sets of numbers. We write x ∈ X if x is an element of the set X and x ∉ X if x is not an element of X. It is convenient to define the empty set, denoted by ∅, as the set with no elements. (Since sets are determined by their elements, there is only one set with no elements!) If X ≠ ∅, meaning that X has at least one element, then we say that X is nonempty. We can define a finite set by listing its elements (between curly brackets). For example, X = {2, 3, 5, 7, 11} is a set with five elements. The order in which the elements are listed or repetitions of the same element are irrelevant. Alternatively, we can define X as the set whose elements are the first five prime numbers. It doesn’t matter how we specify the elements of X, only that they are the same. A function f: X → Y between sets X, Y assigns to each x ∈ X a unique element f(x) ∈ Y. Functions are also called maps, mappings, or transformations. The set X on which f is defined is called the domain of f and the set Y in which it takes its values is called the codomain. We write f: x → f(x) to indicate that f is the function that maps x to f(x). For example, the identity function 𝑖𝑑! : X → X on a set X is the function 𝑖𝑑! : x → x that maps every element to itself. C. Mathematical Logic https://www.karlin.mff.cuni.cz/~krajicek/mendelson.pdf One of the popular definitions of logic is that it is the analysis of methods of reasoning. In studying these methods, logic is interested in the form rather than the content of the argument. For example, consider the two arguments: 1. All men are mortaL Socrates is a man. Hence, Socrates is mortal. 2. All cats like fish. Silvy is a cat. Hence, Silvy likes fish. Both have the same form: All A are B. S is an A. Hence, S is a B. The truth or falsity of the particular premises and conclusions is of no concern to logicians. They want to know only whether the premises imply the conclusion. If the work uses mathematical techniques or if it is primarily devoted to the study of mathematical reasoning, then it may be called mathematical logic. We can narrow the domain of mathematical logic  if we define its principal aim to be a precise and adequate understanding of the notion of mathematical proof. The table below shows the form of compound statements with respect to the logical operation being used. True if and only if P Conjunction 𝑃∧𝑄 P and Q and Q are both true True if and only if P Disjunction 𝑃 ∨𝑄 P or Q is true or Q is true or both are true P implies Q True under all If P then Q circumstances Implication 𝑃→𝑄 Q if P except when P is P only if Q true and Q is false. True if and only if P Bi-conditional 𝑃↔𝑄 P if and only if Q and Q are both true or both false On the other hand, the following table shows the truth values of each compound statement with respect to the logical operation being used: Negation Conjuction Disjunction P ~P P Q 𝑃∧𝑄 P Q 𝑃∨𝑄 T F T T T T T T F T T F F T F T F T F F T T F F F F F F Implication Bi-conditonal P Q 𝑃→𝑄 P Q 𝑃↔𝑄 T T T T T T T F F T F F F T T F T F F F T F F T LEARNING EVALUATION Teaching-Learning Activities TLA1: Language of Mathematics Expected Output: Translation of English Language to Mathematical Form Instruction: Answer each question. 1. Find other names for the number 12 using the following symbols. a. + ______________  b. / ______________ c. √ _____________ 2. Encircle the verb in each sentence and determine whether the statement is True or False. a. EDSA is the longest road in Metro Manila. ___________ b. 8(2) − 6 = √100 ___________ c. 𝜋 < 3 ___________ TLA2: Sets and Functions Expected Output: Sentences to Set Notations Instruction: A. Represent the following sentences to set notations. 1. The set counting numbers greater than -1 and less than 9. 2. The set of integers satisfying 𝑥 − 2 ≤ 5. 3. The set of positive numbers that are divisible by three and less than or equal to 12. B. Determine which is a function from x to y. 4. {(d,a), (a.a), (e,b), (b,c)} 5. {(2,1), (5,2), (5,3), (6,4)} ASSESSMENTS (CO1) AT1. Cat. 2-3: Online Quiz on Mathematical Logic Cat. 1: Answer the following questions, follow instructions for each type of test. A. Formalize the given statements using the following propositions. P: Paul is happy. Q: Queenie is happy. R: Paul plays the guitar. 1. Both Paul and Queenie are happy. 2. Paul plays the guitar provided that he is happy. 3. If Paul is happy and plays the guitar, then Queenie is not happy. B. Construct the truth tables for the given compound statements. 4. 𝑝^(~𝑞) 5. [𝑝^(~𝑞) ]˅[(~𝑝)˅𝑞] 6. ~(𝑝˅~𝑞) ˅𝑝 ASSIGNMENTS (CO1) Conditional Statements (Cat. 1, 2 and 3) Output: Derived Forms of Conditional Statements Instruction: write the converse, inverse and contrapositive of the given statement. Use the contrapositive to determine whether the original statement is true. Note: If the contrapositive statement is true, then the original statement is also true. 1. If yesterday is not Wednesday, then tomorrow is not Friday. Converse: _______________________________________________________________________ Inverse: ________________________________________________________________________ Contrapositive: ___________________________________________________________________ True/False: _______________ RUBRICS FOR GRADING A. Quizzes: Use the numeric scores B. Problem Set 0 1 2 3 4 Student did Student Student Student is Student is not make any attempted to attempted to able to able to attempt to solve 50% of solve all the completely completely solve any of the problems problems in solve 50% of solve 75% of the problems in the problem the problem the problems the problems in the set or set or in the problem in the problem problem set displayed displayed set or set or or prove any logical logical completed completed all of the reasoning reasoning 75% of the the proof/s in statements in 50% of the 75% of the time in the quiz the quiz time in time in attempting to attempting to attempting to prove the prove the prove the statement/s in statement/s in statement/s in the quiz the quiz the quiz REFERENCES Mendelson, E. (1997). Introduction to Mathematical Logic fourth Edition. Retrieved from https://www.karlin.mff.cuni.cz/~krajicek/mendelson.pdf Sets and Functions [PDF]. Retrieved from https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch1.pdf The language of Mathematics [PDF]. Retrieved from http://www.onemathematicalcat.org/pdf_files/LANG1.pdf