MATH(MODULE 6). LEARNING MATERIAL FOR MATH
- 1. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur TINAMBAC CAMPUS . Name of Campus/ College Module #6 Mathematics as a Language Name of Student: _____________________________ Week No: 4 Course Code: GE4 Name of Faculty: MARIVIC S. MORCILLA Course Title: MATHEMATICS IN THE MODERN WORLD I. OBJECTIVES 1. Discuss the language, symbols and conventions of mathematics. 2. Explain the nature of mathematics as a language. 3. Perform operations on mathematical expressions correctly. 4. Acknowledge that mathematics is a useful language. 5. Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts. 6. Write clear and logical proofs. II. LESSON Mathematics Students build a foundation of basic understandings in number, operation, and quantitative reasoning patterns, relationships, and algebraic thinking Geometry and spatial reasoning; measurement; and probability and statistics. They use numbers in ordering, labelling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use objects to create and identify patterns and use those patterns to express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. As they progress, they move from informal to formal language to describe two- and three-dimensional geometric figures and likenesses in the physical world. And begin to associate measurement concepts as they identity and compare attributes of objects and situations. They later collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences. As they widen their scope in mathematics, algorithms are being utilized 1or generalizations. Appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems become their basic tools. Students select and use formal language to describe their reasoning as they Identify, compare, classify and generalize concepts. Mathematics as one progresses, centers on proof, argumentation, personal perspective and insights, convincing people and making them understand. Thus, there is elegance in writing mathematics. It is essential to it. It is like prose and poetry. Since writing mathematically is essential then alter knowing the fundamental concepts, one needs to practice writing. To start with, just like any languages definitions are important. Mathematics holds on to definitions. The concepts needed in mathematics are defined properly. Definition in mathematics is a concise statement. It is concise because it plainly contains the basic properties of an object or concept which unambiguously identify that object or concept. Thus, the essential characteristics of a good definition are concise, basic and unambiguously identified. It is concise and not ramble on with extraneous or unnecessary information. It simply involves basic properties, ideally those that are simply stated and have immediate intuitive appeal. it should not involve properties that require extensive derivation or proof, do t are hard to work with. In order to be complete, a definition must describe exactly the thing being defined; nothing more, and nothing less.
- 2. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur GOOD DEFINITION: A rectangle is a quadrilateral all four of whose angles are right angles. POOR DEFINITION: A rectangle is a parallelogram in which the diagonals have the same length and all the angles are right angles. It can be inscribed in a circle and its area is given by the product of two adjacent sides. This is not CONCISE. It contains too much information, all of which is correct but most of which is unnecessary. POOR DEFINITION: A Rectangle is a parallelogram whose diagonals have equal lengths. This statement is true and concise, but the defining property is not BASIC. This would work better as a theorem to be proved than as a definition. In Mathematics, assertions of this kind are regarded as characterization rather than as definitions. BAD DEFINITION: A rectangle is a quadrilateral with right angles. This is AMBIGUOUS. With some right angles? With all right angles? There are Lots of quadrilaterals that have some right angles but are not rectangles. UNACCEPTABLE DEFINITION: A rectangle has right angles. This is unacceptable because mathematics is written as English is written, in complete, grammatical sentences. Such abbreviations frequently hide major misunderstandings as will be pointed out below. Though mathematics uses a lot of symbols and terminologies, it is not plainly putting them together Just like the English language. There may be sentences that are correct in English language but make no sense in mathematics. Mathematical symbols are a precise form of shorthand. They have to have meaning for you. To help with understanding you have context and convention. In the English language, we have nouns (name given to object of interest) and sentences (those that state a complete thought. Nouns are names of person, place or thing. There are sentences that are true, not true or those which are sometimes true sometimes false. A true sentence in English is "The word 'mathematics' has 11 letters. A false sentence is that "The word mathematics has eight letters.” The sentence “Mathematics is a difficult subject,” is sometimes true, sometimes false. In Mathematical language, there is also an expression (name given to mathematical object of interest) and a mathematical sentence (just like the English language must state a complete thought). Examples of mathematical expressions are number, set, matrix, ordered pair, average. A true mathematical statement is 2 + 3 = 5. An example of a mathematical sentence that is false is that: 2 +3 = 4. A mathematical statement that is sometimes true or sometimes false is: x = 2. Proper writing of mathematical sentences aids to the proper solving of problems and proofs of theorems or conjectures. Expression Mathematical Expression A number increase by 7 x + 7 Thrice a number added to 10 3x + 10 One number is four times the other x, 4x Sum of three consecutive integers x + (x + 1) + (x + 2) Ten less than four times a certain number 4x – 10 mother is 6 years more than three times older than her son If x is son's age, then, three times son's age is 3x and six more than 3x is 3x + 6 LOGIC AND REASONING Mathematics started as a practical technique to immediate problems in life. This was compiled and has been applicable in governance. In the hands of the ancient Greeks mathematics becomes a systematic body of
- 3. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur knowledge. Mathematics is established as a deductive science in which the standard of rigorous demonstration is deductive proof. Aristotle provided a codification of logic which remains definitive for two thousand years. The axiomatic method is established and is systematically applied to the mathematics of the classical period by Euclid, whose Elements becomes one of the most influential books in history. The next major advances in logic after Aristotle appear in the nineteenth century, in which Boole introduces the propositional (Boolean) logic and Frege devises the predicate calculus. His provides the technical basis for the logicization of mathematics and the transition from informal to formal proof. There is abundant list of statements in mathematics. These statements differ from the statements in communication. Some statements are best confirmed by argument. If I were to tell you, while sitting in a confined room with no windows, that it is raining outside right now then there is no amount of argument that would be as convincing as stepping outside to see for yourself. The statement "it is raining is not an analytical statement about the relationship between concepts. It is but a synthetic proposition about the world that might or might not be true at any given time. In contrast, mathematical statements are analytical statements that are better proved argument than by experiment. This sets the difference of mathematics from the other sciences. Reasoning in mathematics is different. Logic is the science of reasoning, proof, thinking, or inference. Logic allows analyze a piece of deductive reasoning and determine whether it is correct or not – to determine if the argument is valid or invalid. It is a tool used in mathematical proofs. The rules of logic specify the meaning of mathematical statements. For instance, these rules help us understand and reason with statements. Logic is the basis of all mathematical reasoning, and of all automated reasoning. Logic is a tool for working with complicated compound statements. It includes: • A language for expressing them. • A concise notation for writing them • A methodology for objectively reasoning about their truth or Falsity. • It is the foundation for expressing formal proofs in all branches of mathematics. A mathematical proof is an argument that begins with a set of postulates or assumptions and proceeds to a conclusion by agreed methods of argument. A deductive argument is one that, if valid, has a conclusion that is entailed by its premises. Meaning, the truth of the conclusion is a logical consequence of the premises – if the premises are true, then the conclusion must be true. A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. As a declarative sentence, it expresses a complete thought with a definite meaning. Examples: 1. Saint Louis University is in Baguio City. 2. Quezon City is the capital of the Philippines. 3. Benguet is a part of the Cordillera Administrative Region. 4. 1 + 1 = 2 5. 2 + 2 = 3. Not all sentences are considered propositions. For instance, consider the following: 1. What day is today? 2. Read the instructions carefully. 3. 5x + 2 = 3. 4. x+ 2y = 3z. Sentences 1 and 2 are not propositions since they are not declarative sentences; sentences 3 and 4 are not propositions because they are neither true nor false; sentences 3 and 4 can be turned into a proposition if we assign values to the variables.
- 4. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur Mathematically the propositions are converted symbolically. We use letters to denote propositional variables (or statement variables), that is, variables that represent propositions, just as letters are used to denote numerical variables. The conventional letters used for propositional variables are P, Q, R, S, . . . . The truth value of a proposition is true, denoted by T, if is a true proposition, and the truth value of a proposition is false, denoted by F, if it is false proposition. Many mathematical statements are constructed by combining one or more propositions. New propositions, called compound propositions, are formed from existing propositions using logical operators or logical connectives. Logical Connective is a word or symbol that joins two sentences or propositions to produce new one. The five basic logical connectives are conjunction, disjunction, implication, bi-conditional, and negation. The table below shows the different logical connectives and the corresponding Key words used and the symbol used. Name Connective (Key Word) Symbol Conjunction And Ʌ Disjunction Or V Implication or Conditional If… then… Biconditional … if and only if… Negation Not ~ or ¬ Conditional statements play an essential role in mathematical reasoning. They are often seen in the different theorems of mathematics. There are different ways of expressing p → q. The common ones encountered in mathematics are: "p implies q, "if p, then q,” "if p, q”, “p is sufficient for q”, “q if p”, “q when p”, "a necessary condition for p is q”, “q unless not p", "p only if q”, "a sufficient condition for q is p”, “q whenever p”, “q is necessary for p”, and “q follows from p”. Negation of mathematical statement P is denoted by ¬P, read as “not P”. If P is true, then ¬ P is not true. Examples: 1. P: The trainees are sleepy. ¬ P: The trainees are not sleepy. 2. Q: I have a new phone. ¬Q: I do not have a new phone. TRUTH VALUES Summary of truth values of compound statements using logical connectives. The conditional statement can be transformed to new conditional statements. In particular, there are three related conditional statements that occur so often that they are special names. These are the converse, the contrapositive and the inverse. The contrapositive always has the same truth value as the conditional. Assigning P as the antecedent or hypothesis and Q as the consequent or conclusion; the conditional and the implications or transformation of it is as follows:
- 5. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur When two compound propositions always have the same truth value, we call the equivalent, so that a conditional statement and its contrapositive are equivalent. The converse and the inverse of a conditional statement are equivalent statements but neither is equivalent to the original conditional statement. An important thing to remember is that one of the most common logical errors is to assume that the converse or the inverse of a conditional statement is equivalent to this conditional statement. A deductive argument is one that, if valid, has a conclusion that is entailed by its premises. Meaning, the truth of the conclusion is a logical consequence of the premises – if the premises are true, then the conclusion must be true. A logical argument has three stages: • Premises: The claims that are given in support of an argument – the building blocks of a logical argument. A premise is a propositional statement which is either true or false. • Inference: The logical move from one or more premises to arrive at its conclusion. All inferences must abide by a rule of inference for an argument to be valid. • Conclusion: The premise that is the consequence, or product, of the above premises + inference.* A conclusion can then itself become premise (building block) of a continued or new argument. Symbolic logic symbolizes arguments tor simple, efficient assessment of validity. As arguments get longer and more complex, symbols are especially important. Writing an Argument in Symbolic Form Write Arguments in Symbolic Form and Valid Arguments Given propositions: I have a college degree (p) I am lazy (q) Using the propositions and creating an argument: I have a college degree, then am not lazy. I don't have a college degree. Therefore, l am lazy. Symbolic form: Therefore, this argument is invalid because the last column has a false item.
- 6. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur Illustrative Example: Symbolize the argument, construct a truth table, and determine if the argument is valid. It will be sunny or cloudy today. It isn’t sunny. Therefore, it will be cloudy. S = It will be sunny C = It will be cloudy This is a valid argument. Rules of Inference Illustrative Example Show that the premises: “It is not sunny this afternoon and it is colder than yesterday,” “We will go swimming only if it is sunny,” “If we do not go swimming, then we will take a canoe trip," and “If we take a canoe trip, then we will be home by sunset" Lead to the conclusion: "We will be home by sunset." Let: P be the proposition: “It is sunny this afternoon." Q be the proposition: “It is colder than yesterday." R be the proposition: “We will go swimming." S be the proposition: “We will take a canoe trip.” T be the proposition: "we will be home by sunset."
- 7. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur Constructing an argument to show that the premises lead to the desired conclusion: STEP: REASON: Note: A truth table could be used in the proof of the conclusion, that is to show that each of the four hypotheses are true, the conclusion is also true. However, since there are five propositional variables, p, q, r, s, and t, the truth table would have 25 rows or 32 rows. When the variables in a propositional function are assigned values, the resulting statement becomes a proposition with a certain truth value. However, there is another important way, called quantification. To create a proposition from a propositional function. It expresses the words all, some, many, none and few are used in quantification. A quantifier is a constructs that specifies the quantity of specimens in the domain of discourse that satisfy the formula. TWO TYPES OF QUANTIFICATION
- 8. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur Negating Quantified Expressions
- 9. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur Illustrative Examples: III. ACTIVITIES. Fill in the blanks.
- 10. Republic of the Philippines PARTIDO STATE UNIVERSITY Goa, Camarines Sur IV. ASSESSMENT. Determine the truth value of all the statements. Write True of False. V. SUGGESTED REFERENCES 1. Aufmann. Richard et. Al. Mathematics in the Modern World, 2018, REX Book Store Inc. 2. Sirug, Winston Mathematics of Investment, 3. Cabero et.al. Quantitative Techniques in Management, 2005, National Book Store 4. Aufmann et. al.Mathematical Excursion (4th Ed) 5. Guillermo, Raflyn M. Mathematics in the Modern World A Worktext, 2018, Nieme Publishing House Co. Ltd.