Mathematics for Natural Sciences PDF

MINISTRY OF SCIENCE AND HIGHER EDUCATION ashkas Mathematics for Natural Sciences Prepared by: 1. Ato Arbsie Yasin 2. Dr. Berhanu Bekele 3. Dr. Berhanu Guta 4. Ato Wondwosen Zemene MOSHE SEPTEMBER, 2019  Content...

MINISTRY OF SCIENCE AND HIGHER EDUCATION ashkas Mathematics for Natural Sciences Prepared by: 1. Ato Arbsie Yasin 2. Dr. Berhanu Bekele 3. Dr. Berhanu Guta 4. Ato Wondwosen Zemene MOSHE SEPTEMBER, 2019  Content Page Chapter 1: Propositional Logic and set Theory ………………………………………….. 1 1.1. Propositional Logic ……………………………………………… 1 1.1.1 Definition and examples of Propositions ……………………… 2 1.1.2 Logical connectives ……………………………………………. 2 1.1.3 Compound (or complex) proposition ………………………….. 7 1.1.4 Tautology and contradiction …………………………………… 10 1.2. Open propositions and quantifiers ……………………………………... 12 1.3. Arguments and Validity ………………………………………………... 20 1.4. Set Theory …………………………………………………………. 25 1.4.1 The Concept of a set …………………………………………….. 26 1.4.2 Description of sets ………………………………………………. 26 1.4.3 Set operations and Venn diagrams ……………………………… 31 Chapter 2: The Real and Complex Number Systems ……………………………………... 39 2.1 The real number system ………………………………………………... 39 2.1.1 The natural numbers, principle of mathematical induction and the well ordering axiom ………………………………… 39 2.1.2 The set of integers ……………………………………………. 45 2.1.3 The set of rational numbers ………………………………….. 48 2.1.4 The set of real numbers, upper bound and lower bound, least Upper bound and greatest lower bound; completeness property of real numbers ………………………………………………. 52 2.2 The set of complex numbers...………………………………………….. 56 2.2.1 Plotting complex numbers…………………………………….. 57 2.2.2 Operations on complex numbers ………………………………57 2.2.3 Conjugate of a complex number ……………………………… 58 2.2.4 Modulus (Norm) of a complex number ………………………. 59 2.2.5 Additive and multiplicative inverse …………………………... 60 2.2.6 Argument of a complex number ……………………………….61 2.2.7 Polar form of a complex numbers................…………………. 62 2.2.8 Extraction of roots ……………………………………………. 66 Chapter 3: Functions ……………………………………………………………………… 68 3.1 Review of relations and functions ……………………………………... 68 3.2 Real valued functions and their properties …………………………….. 76 3.3 Types of functions and inverse of a function ………………………….. 81 3.4 Polynomials, zeros of polynomials, rational functions and their graphs ……………………………………………………………. 85 3.5 Definition and basic properties of logarithmic, exponential, trigonometric and hyperbolic functions, and their graphs. ……………. 97 Chapter 4: Analytic Geometry ……………………………………………………………. 123 4.1 Distance Formula and Equation of Lines ……………………………. 124 4.1.1 Distance between two points and division of segments …….124 4.1.2 Equations of lines …………………………………………… 127 4.1.3 Distance between a point and a line …………………………131 4.2 Circles ………………………………………………………………...... 132 4.2.1 Definition of a circle …………………………………………. 133 4.2.2 Equation of a circle …………………………………………...... 134 4.2.3 Intersection of a circle with a line and tangent line to a circle... 137 4.3 Parabolas …………………………………………………………………. 139 4.3.1 Definition of parabola ………………………………………….. 140 4.3.2 Equation of parabolas …………………………………………... 141 4.4 Ellipse …………………………………………………………………….. 146 4.4.1 Definition of ellipse …………………………………………….. 146 4.4.2 Equation of ellipse ……………………………………………… 148 4.5 Hyperbola ………………………………………………………………… 152 4.5.1 Definition of a hyperbola ……………………………………...... 153 4.5.2 Equation of a hyperbola ………………………………………… 155 4.6 The general second degree equation ……………………………………… 162 4.6.1 Rotation of coordinate axes …………………………………….. 163 4.6.2 Analysis of the general second degree equations..……………... 166 References …………………………………………………………………………………… 171  Chapter One Propositional Logic and Set Theory In this chapter, we study the basic concepts of propositional logic and some part of set theory. In the first part, we deal about propositional logic, logical connectives, quantifiers and arguments. In the second part, we turn our attention to set theory and discus about description of sets and operations of sets. Main Objectives of this Chapter At the end of this chapter, students will be able to:-  Know the basic concepts of mathematical logic.  Know methods and procedures in combining the validity of statements.  Understand the concept of quantifiers.  Know basic facts about argument and validity.  Understand the concept of set.  Apply rules of operations on sets to find the result.  Show set operations using Venn diagrams. 1.1. Propositional Logic Mathematical or symbolic logic is an analytical theory of the art of reasoning whose goal is to systematize and codify principles of valid reasoning. It has emerged from a study of the use of language in argument and persuasion and is based on the identification and examination of those parts of language which are essential for these purposes. It is formal in the sense that it lacks reference to meaning. Thereby it achieves versatility: it may be used to judge the correctness of a chain of reasoning (in particular, a "mathematical proof") solely on the basis of the form (and not the content) of the sequence of statements which make up the chain. There is a variety of symbolic logics. We shall be concerned only with that one which encompasses most of the deductions of the sort encountered in mathematics. Within the context of logic itself, this is "classical" symbolic logic. Section objectives: After completing this section, students will be able to:-  Identify the difference between proposition and sentence.  Describe the five logical connectives.  Determine the truth values of propositions using the rules of logical connectives. 1   Construct compound propositions using the five logical connectives.  Identify the difference between the converse and contrapositive of conditional statements.  Determine the truth values of compound propositions.  Distinguish a given compound proposition is whether tautology or contradiction. 1.1.1. Definition and examples of propositions Consider the following sentences. a. 2 is an even number. b. A triangle has four sides. c. Athlete Haile G/silassie weighed 45 kg when he was 20 years old. d. May God bless you! e. Give me that book. f. What is your name? The first three sentences are declarative sentences. The first one is true and the second one is false. The truth value of the third sentence cannot be ascertained because of lack of historical records but it is, by its very form, either true or false but not both. On the other hand, the last three sentences have no truth value. So they are not declaratives. Now we begin by examining proposition, the building blocks of every argument. A proposition is a sentence that may be asserted or denied. Proposition in this way are different from questions, commands, and exclamations. Neither questions, which can be asked, nor exclamations, which can be uttered, can possibly be asserted or denied. Only propositions assert that something is (or is not) the case, and therefore only they can be true or false. Definition 1.1: A proposition (or statement) is a sentence which has a truth value (either True or False but not both). The above definition does not mean that we must always know what the truth value is. For example, the sentence “The 1000th digit in the decimal expansion of is 7” is a proposition, but it may be necessary to find this information in a Web site on the Internet to determine whether this statement is true. Indeed, for a sentence to be a proposition (or a statement), it is not a requirement that we are able to determine its truth value. Every proposition has a truth value, namely true (denoted by ) or false (denoted by ). 1.1.2. Logical connectives In mathematical discourse and elsewhere one constantly encounters declarative sentences which have been formed by modifying a statement with the word “not” or by connecting statements with the words “and”, “or”, “if... then (or implies)”, and “if and only if”. These five words or combinations of words are called propositional connectives. Note: Letters such as etc. are usually used to denote propositions. 2 Conjunction When two propositions are joined with the connective “and,” the proposition formed is a logical conjunction. “and” is denoted by “ ”. So, the logical conjunction of two propositions, and , is written: , read as “ and ,” or “ conjunction ”. p and q are called the components of the conjunction. is true if and only if is true and is true. The truth table for conjunction is given as follows: Example 1.1: Consider the following propositions: : 3 is an odd number. (True) : 27 is a prime number. (False) : Addis Ababa is the capital city of Ethiopia. (True) a. : 3 is an odd number and 27 is a prime number. (False) b. : 3 is an odd number and Addis Ababa is the capital city of Ethiopia. (True) Disjunction When two propositions are joined with the connective “or,” the proposition formed is called a logical disjunction. “or” is denoted by “ ”. So, the logical disjunction of two propositions, and , is written: read as “ or ” or “ disjunction.” is false if and only if both and are false. The truth table for disjunction is given as follows: 3 Example 1.2: Consider the following propositions: : 3 is an odd number. (True) : 27 is a prime number. (False) : Nairobi is the capital city of Ethiopia. (False) a. : 3 is an odd number or 27 is a prime number. (True) b. : 27 is a prime number or Nairobi is the capital city of Ethiopia. (False) Note: The use of “or” in propositional logic is rather different from its normal use in the English language. For example, if Solomon says, “I will go to the football match in the afternoon or I will go to the cinema in the afternoon,” he means he will do one thing or the other, but not both. Here “or” is used in the exclusive sense. But in propositional logic, “or” is used in the inclusive sense; that is, we allow Solomon the possibility of doing both things without him being inconsistent. Implication When two propositions are joined with the connective “implies,” the proposition formed is called a logical implication. “implies” is denoted by “.” So, the logical implication of two propositions, and , is written: read as “ implies.” The function of the connective “implies” between two propositions is the same as the use of “If … then …” Thus can be read as “if , then.” is false if and only if is true and is false. This form of a proposition is common in mathematics. The proposition is called the hypothesis or the antecedent of the conditional proposition while is called its conclusion or the consequent. The following is the truth table for implication. Examples 1.3: Consider the following propositions: : 3 is an odd number. (True) : 27 is a prime number. (False) : Addis Ababa is the capital city of Ethiopia. (True) : If 3 is an odd number, then 27 is prime. (False) : If 3 is an odd number, then Addis Ababa is the capital city of Ethiopia. (True) 4 We have already mentioned that can be expressed as both “If , then ” and “ implies. ” There are various ways of expressing the proposition , namely: If , then. if. implies. only if. is sufficient for. is necessary for Bi-implication When two propositions are joined with the connective “bi-implication,” the proposition formed is called a logical bi-implication or a logical equivalence. A bi-implication is denoted by “ ”. So the logical bi- implication of two propositions, and , is written:. is false if and only if and have different truth values. The truth table for bi-implication is given by: Examples 1.4: a. Let : 2 is greater than 3. (False) : 5 is greater than 4. (True) Then : 2 is greater than 3 if and only if 5 is greater than 4. (False) b. Consider the following propositions: : 3 is an odd number. (True) : 2 is a prime number. (True) : 3 is an odd number if and only if 2 is a prime number. (True) There are various ways of stating the proposition. if and only if (also written as iff ), implies and implies , is necessary and sufficient for is necessary and sufficient for 5  is equivalent to Negation Given any proposition , we can form the proposition called the negation of. The truth value of is if is and if is. We can describe the relation between and as follows. Example 1.5: Let : Addis Ababa is the capital city of Ethiopia. (True) : Addis Ababa is not the capital city of Ethiopia. (False) Exercises 1. Which of the following sentences are propositions? For those that are, indicate the truth value. a. 123 is a prime number. b. 0 is an even number. c.. d. Multiply by 3. e. What an impossible question! 2. State the negation of each of the following statements. a. is a rational number. b. 0 is not a negative integer. c. 111 is a prime number. 3. Let : 15 is an odd number. : 21 is a prime number. State each of the following in words, and determine the truth value of each. a.. e.. b.. f.. c.. a.. d.. g.. 6  4. Complete the following truth table. 1.1.3. Compound (or complex) propositions So far, what we have done is simply to define the logical connectives, and express them through algebraic symbols. Now we shall learn how to form propositions involving more than one connective, and how to determine the truth values of such propositions. Definition 1.2: The proposition formed by joining two or more proposition by connective(s) is called a compound statement. Note: We must be careful to insert the brackets in proper places, just as we do in arithmetic. For example, the expression will be meaningless unless we know which connective should apply first. It could mean or , which are very different propositions. The truth value of such complicated propositions is determined by systematic applications of the rules for the connectives. The possible truth values of a proposition are often listed in a table, called a truth table. If and are propositions, then there are four possible combinations of truth values for and. That is, , , and. If a third proposition is involved, then there are eight possible combinations of truth values for , and. In general, a truth table involving “ ” propositions , ,…, contains possible combinations of truth values. So, we use truth tables to determine the truth value of a compound proposition based on the truth value of its constituent component propositions. Examples 1.6: a. Suppose and are true and and are false. What is the truth value of ? i. Since is true and is false, is false. ii. Since is true and is false, is true. iii. Thus by applying the rule of implication, we get that is true. b. Suppose that a compound proposition is symbolized by 7 and that the truth values of and are and , respectively. Then the truth value of is , that of is , that of is. So the truth value of is. Remark: When dealing with compound propositions, we shall adopt the following convention on the use of parenthesis. Whenever “ ” or “ ” occur with “ ” or “ ”, we shall assume that “ ” or “ ” is applied first, and then “ ” or “ ” is then applied. For example, means means means means However, it is always advisable to use brackets to indicate the order of the desired operations.. Definition 1.3: Two compound propositions and are said to be equivalent if they have the same truth value for all possible combinations of truth values for the component propositions occurring in both and. In this case we write. Example 1.7: Let.. Then, is equivalent to , since columns 5 and 6 of the above table are identical. Example 1.8: Let.. Then Looking at columns 5 and 6 of the table we see that they are not identical. Thus. It is useful at this point to mention the non-equivalence of certain conditional propositions. Given the conditional , we give the related conditional propositions:- 8  : Converse of : Inverse of : Contrapositive of As we observed from example 1.7, the conditional and its contrapositve are equivalent. On the other hand, and. Do not confuse the contrapositive and the converse of the conditional proposition. Here is the difference: Converse: The hypothesis of a converse statement is the conclusion of the conditional statement and the conclusion of the converse statement is the hypothesis of the conditional statement. Contrapositive: The hypothesis of a contrapositive statement is the negation of conclusion of the conditional statement and the conclusion of the contrapositive statement is the negation of hypothesis of the conditional statement. Example 1.9: a. If Kidist lives in Addis Ababa, then she lives in Ethiopia. Converse: If Kidist lives in Ethiopia, then she lives in Addis Ababa. Contrapositive: If Kidist does not live in Ethiopia, then she does not live in Addis Ababa. Inverse: If Kidist does not live in Addis Ababa, then she does not live in Ethiopia. b. If it is morning, then the sun is in the east. Converse: If the sun is in the east, then it is morning. Contrapositive: If the sun is not in the east, then it is not morning. Inverse: If it is not morning, then the sun is not the east. Propositions, under the relation of logical equivalence, satisfy various laws or identities, which are listed below. 1. Idempotent Laws a.. b.. 2. Commutative Laws a.. b.. 3. Associative Laws a.. b.. 4. Distributive Laws a.. b.. 5. De Morgan’s Laws 9  a.. b. 6. Law of Contrapositive 7. Complement Law. 1.1.4. Tautology and contradiction Definition: A compound proposition is a tautology if it is always true regardless of the truth values of its component propositions. If, on the other hand, a compound proposition is always false regardless of its component propositions, we say that such a proposition is a contradiction. Note: A proposition that is neither a tautology nor a contradiction is called a contingency. Examples 1.10: a. Suppose is any proposition. Consider the compound propositions and. Observe that is a tautology while is a contradiction. b. For any propositions and. Consider the compound proposition. Let us make a truth table and study the situation. T T T T We have exhibited all the possibilities and we see that for all truth values of the constituent propositions, the proposition is always true. Thus, is a tautology. c. The truth table for the compound proposition. 10 In example 1.10(c), the given compound proposition has a truth value for every possible combination of assignments of truth values for the component propositions and. Thus is a contradiction. Remark: 1. In a truth table, if a proposition is a tautology, then every line in its column has as its entry; if a proposition is a contradiction, every line in its column has as its entry. 2. Two compound propositions and are equivalent if and only if “ ” is a tautology. Exercises 1. For statements and , use a truth table to show that each of the following pairs of statements are logically equivalent. a. and. b. and. c. and. d. and. e. and. 2. For statements , and , show that the following compound statements are tautology. a.. b.. c.. 3. For statements and , show that is a contradiction. 4. Write the contrapositive and the converse of the following conditional statements. a. If it is cold, then the lake is frozen. b. If Solomon is healthy, then he is happy. c. If it rains, Tigist does not take a walk. 5. Let and be statements. Which of the following implies that is false? a. is false. d. is true. b. is true. e. is false. c. is true. 6. Suppose that the statements and are assigned the truth values and , respectively. Find the truth value of each of the following statements. f.. a.. g.. b.. h.. c.. 11  d.. i.. e.. j.. 7. Suppose the value of is ; what can be said about the value of ? 8. a. Suppose the value of is ; what can be said about the values of and ? b. Suppose the value of is ; what can be said about the values of and ? 9. Construct the truth table for each of the following statements. a.. d.. b.. e.. c.. f.. 10. For each of the following determine whether the information given is sufficient to decide the truth value of the statement. If the information is enough, state the truth value. If it is insufficient, show that both truth values are possible. a. , where. b. , where. c. , where. d. , where. e. , where. f. , where and. 1.2. Open propositions and quantifiers In mathematics, one frequently comes across sentences that involve a variable. For example, is one such. The truth value of this statement depends on the value we assign for the variable. For example, if , then this sentence is true, whereas if , then the sentence is false. Section objectives: After completing this section, students will be able to:-  Define open proposition.  Explain and exemplify the difference between proposition and open proposition.  Identify the two types of quantifiers.  Convert open propositions into propositions using quantifiers.  Determine the truth value of a quantified proposition. 12   Convert a quantified proposition into words and vise versa.  Explain the relationship between existential and universal quantifiers.  Analyze quantifiers occurring in combinations. Definition 1.4: An open statement (also called a predicate) is a sentence that contains one or more variables and whose truth value depends on the values assigned for the variables. We represent an open statement by a capital letter followed by the variable(s) in parenthesis, e.g., etc. Example 1.11: Here are some open propositions: a. is the day before Sunday. b. is a city in Africa. c. is greater than. d.. It is clear that each one of these examples involves variables, but is not a proposition as we cannot assign a truth value to it. However, if individuals are substituted for the variables, then each one of them is a proposition or statement. For example, we may have the following. a. Monday is the day before Sunday. b. London is a city in Africa. c. 5 is greater than 9. d. –13 + 4= –9 Remark The collection of all allowable values for the variable in an open sentence is called the universal set (the universe of discourse) and denoted by. Definition 1.5: Two open proposition and are said to be equivalent if and only if for all individual. Note that if the universe is specified, then and are equivalent if and only if for all. Example 1.12: Let.. Let. Then for all ; and have the same truth value. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Therefore for all. 13 Definition 1.6: Let be the universal set. An open proposition is a tautology if and only if is always true for all values of. Example 1.13: The open proposition is a tautology. As we have observed in example 1.11, an open proposition can be converted into a proposition by substituting the individuals for the variables. However, there are other ways that an open proposition can be converted into a proposition, namely by a method called quantification. Let be an open proposition over the domain. Adding the phrase “For every ” to or “For some ” to produces a statement called a quantified statement. Consider the following open propositions with universe. a.. b.. c.. Then is always true for each , is true only for and , is always false for all values of. Hence, given an open proposition , with universe , we observe that there are three possibilities. a. is true for all. b. is true for some. c. is false for all. Now we proceed to study open propositions which are satisfied by “all” and “some” members of the given universe. a. The phrase "for every " is called a universal quantifier. We regard "for every ," "for all ," and "for each " as having the same meaning and symbolize each by “.” Think of the symbol as an inverted (representing all). If is an open proposition with universe , then is a quantified proposition and is read as “every has the property.” b. The phrase "there exists an " is called an existential quantifier. We regard "there exists an ," "for some ," and "for at least one " as having the same meaning, and symbolize each by “.” Think of the symbol as the backwards capital (representing exists). If is an open proposition with universe , then is a quantified proposition and is read as “there exists with the property.” Remarks: 14  i. To show that is , it is sufficient to find at least one such that is. Such an element is called a counter example. ii. is if we cannot find any having the property. Example 1.14: a. Write the following statements using quantifiers. i. For each real number. Solution:. ii. There is a real number such that. Solution:. iii. The square of any real number is nonnegative. Solution:. b. i. Let. The truth value for [i.e ] is. ii. Let. The truth value for is. is a counterexample since but. On the other hand, is true, since such that. iii. Let. The truth value for is since there is no real number whose absolute value is. Relationship between the existential and universal quantifiers If is a formula in , consider the following four statements. a.. b.. c.. d.. We might translate these into words as follows. a. Everything has property. b. Something has property. c. Nothing has property. d. Something does not have property. Now (d) is the denial of (a), and (c) is the denial of (b), on the basis of everyday meaning. Thus, for example, the existential quantifier may be defined in terms of the universal quantifier. Now we proceed to discuss the negation of quantifiers. Let be an open proposition. Then is false only if we can find an individual “ ” in the universe such that is false. If we succeed in getting such an individual, then is true. Hence will be false if is true. Therefore the negation of is. Hence we conclude that 15. Similarly, we can easily verified that. Remark: To negate a statement that involves the quantifiers and , change each to , change each to , and negate the open statement. Example 1.15: Let. a.. b.. Given propositions containing quantifiers we can form a compound proposition by joining them with connectives in the same way we form a compound proposition without quantifiers. For example, if we have and we can form. Consider the following statements involving quantifiers. Illustrations of these along with translations appear below. a. All rationals are reals.. b. No rationals are reals.. c. Some rationals are reals.. d. Some rationals are not reals.. Example 1.16: Let The set of integers. Let : is a prime number. : is an even number. : is an odd number. Then a. is ; since there is an , say 2, such that is. b. is. As a counterexample take 7. Then is and is. Hence. c. is. d. is. Quantifiers Occurring in Combinations So far, we have only considered cases in which universal and existential quantifiers appear simply. However, if we consider cases in which universal and existential quantifiers occur in combination, we are lead to essentially new logical structures. The following are the simplest forms of combinations: 16  1. “for all and for all the relation holds”; 2. “there is an and there is a for which holds”; 3. “for every there is a such that holds”; 4. “there is an which stands to every in the relation.” Example 1.17: Let The set of integers. Let. a. means that there is an integer and there is an integer such that. This statement is true when and , since 4 + 1 = 5. Therefore, the statement is always true for this universe. There are other choices of and for which it would be true, but the symbolic statement merely says that there is at least one choice for and which will make the statement true, and we have demonstrated one such choice. b. means that there is an integer such that for every ,. This is false since no fixed value of will make this true for all in the universe; e.g. if , then is false for some. c. means that for every integer , there is an integer such that. Let , then will always be an integer, so this is a true statement. d. means that for every integer and for every integer ,. This is false, for if and , we get. Example 1.18: a. Consider the statement For every two real numbers and ,. If we let where the domain of both and is , the statement can be expressed as or as. 17 Since and for all real numbers and , it follows that and so is true for all real numbers and. Thus the quantified statement is true. b. Consider the open statement where the domain of the variable is the set of even integers and the domain of the variable is the set of odd integers. Then the quantified statement can be expressed in words as There exist an even integer and an odd integer such that. Since is true, the quantified statement is true. c. Consider the open statement where the domain of both and is the set of positive rational numbers. Then the quantified statement can be expressed in words as For every positive rational number , there exists a positive rational number such that. It turns out that the quantified statement is true. If we replace by , then we have. Since and for every real number , is false. d. Consider the open statement is odd where the domain of both and is the set of natural numbers. Then the quantified statement , expressed in words, is There exists a natural number such that for every natural numbers , is odd. The statement is false. In general, from the meaning of the universal quantifier it follows that in an expression the two universal quantifiers may be interchanged without altering the sense of the sentence. This also holds for the existential quantifies in an expression such as. In the statement , the choice of is allowed to depend on - the that works for one need not work for another. On the other hand, in the statement , the must work for all , i.e., is independent of. For example, the expression , where and are variables referring to the domain of real numbers, constitutes a true proposition, namely, “For every number , there is a number , such that is less that ,” i.e., “given any number, there is a greater number.” However, if the order of the symbol and is changed, in this case, we obtain: , which is a false proposition, namely, 18 “There is a number which is greater than every number.” By transposing and , therefore, we get a different statement. The logical situation here is:. Finally, we conclude this section with the remark that there are no mechanical rules for translating sentences from English into the logical notation which has been introduced. In every case one must first decide on the meaning of the English sentence and then attempt to convey that same meaning in terms of predicates, quantifiers, and, possibly, individual constants. Exercises 1. In each of the following, two open statements and are given, where the domain of both and is. Determine the truth value of for the given values of and. a.. and.. b.. and.. c.. and.. 2. Let denote the set of odd integers and let is even, and is even. be open statements over the domain. State and in words. 3. State the negation of the following quantified statements. a. For every rational number , the number is rational. b. There exists a rational number such that. 4. Let is an integer. be an open sentence over the domain. Determine, with explanations, whether the following statements are true or false: a.. b.. 5. Determine the truth value of the following statements. a.. b.. c.. d.. e.. f.. g.. h. 6. Consider the quantified statement For every and , is prime. 19  where the domain of the variables and is. a. Express this quantified statement in symbols. b. Is the quantified statement in (a) true or false? Explain. c. Express the negation of the quantified statement in (a) in symbols. d. Is the negation of the quantified in (a) true or false? Explain. 7. Consider the open statement where the domain of is and the domain of is. a. State the quantified statement in words. b. Show quantified statement in (a) is true. 8. Consider the open statement where the domain of is and the domain of is. a. State the quantified statement in words. b. Show quantified statement in (a) is true. 1. 3. Argument and Validity Section objectives: After completing this section, students will be able to:-  Define argument (or logical deduction).  Identify hypothesis and conclusion of a given argument.  Determine the validity of an argument using a truth table.  Determine the validity of an argument using rules of inferences. Definition 1.7: An argument (logical deduction) is an assertion that a given set of statements , called hypotheses or premises, yield another statement , called the conclusion. Such a logical deduction is denoted by: or  Example 1.19: Consider the following argument: If you study hard, then you will pass the exam. You did not pass the exam. 20  Therefore, you did not study hard. Let : You study hard. : You will pass the exam. The argument form can be written as: pq q p When is an argument form accepted to be correct? In normal usage, we use an argument in order to demonstrate that a certain conclusion follows from known premises. Therefore, we shall require that under any assignment of truth values to the statements appearing, if the premises became all true, then the conclusion must also become true. Hence, we state the following definition. Definition 1.8: An argument form is said to be valid if is true whenever all the premises are true; otherwise it is invalid. Example 1.20: Investigate the validity of the following argument: a. p  q,  q   p b. p  q,  q  r  p c. If it rains, crops will be good. It did not rain. Therefore, crops were not good. Solution: First we construct a truth table for the statements appearing in the argument forms. a. The premises and are true simultaneously in row 4 only. Since in this case is also true, the argument is valid. b. 21 The 1st, 2nd, 5th, 6th and 7th rows are those in which all the premises take value. In the 5th, 6th and 7th rows however the conclusion takes value. Hence, the argument form is invalid. c. Let : It rains. : Crops are good. : It did not rain. : Crops were not good. The argument form is Now we can use truth table to test validity as follows: The premises and are true simultaneously in row 4 only. Since in this case is also true, the argument is valid. Remark: 1. What is important in validity is the form of the argument rather than the meaning or content of the statements involved. 2. The argument form is valid iff the statement is a tautology. Rules of inferences Below we list certain valid deductions called rules of inferences. 1. Modes Ponens 2. Modes Tollens 3. Principle of Syllogism 22  4. Principle of Adjunction a. b. 5. Principle of Detachment 6. Modes Tollendo Ponens 7. Modes Ponendo Tollens 8. Constructive Dilemma 9. Principle of Equivalence 10. Principle of Conditionalization Formal proof of validity of an argument Definition 1.9: A formal proof of a conclusion given hypotheses is a sequence of stapes, each of which applies some inference rule to hypotheses or previously proven statements (antecedent) to yield a new true statement (the consequent). 23 A formal proof of validity is given by writing on the premises and the statements which follows from them in a single column, and setting off in another column, to the right of each statement, its justification. It is convenient to list all the premises first. Example 1.21: Show that is valid. Solution: 1. is true premise 2. is true premise 3. is true contrapositive of (2) 4. is true Modes Ponens using (1) and (3) Example 1.22: Show that the hypotheses It is not sunny this afternoon and it is colder than yesterday. If we go swimming, then it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip, then we will be home by sunset. Lead to the conclusion: We will be home by sunset. Let : It is sunny this afternoon. : It is colder than yesterday. : We go swimming. : We take a canoe trip. : We will be home by sunset. Then 1. hypothesis 2. simplification using (1) 3. hypothesis 4. Modus Tollens using (2) and (3) 5. hypothesis 6. Modus Ponens using (4) and (5) 7. hypothesis 8. Modus Ponens using (6) and (7) Exercises 1. Use the truth table method to show that the following argument forms are valid. i.. ii.. iii.. iv.. 24  v.. 2. For the following argument given a, b and c below: i. Identify the premises. ii. Write argument forms. iii. Check the validity. a. If he studies medicine, he will get a good job. If he gets a good job, he will get a good wage. He did not get a good wage. Therefore, he did not study medicine. b. If the team is late, then it cannot play the game. If the referee is here, then the team is can play the game. The team is late. Therefore, the referee is not here. c. If the professor offers chocolate for an answer, you answer the professor’s question. The professor offers chocolate for an answer. Therefore, you answer the professor’s question 3. Give formal proof to show that the following argument forms are valid. a.. b.. c.. d.. e.. f.. g.. h.. i.. 4. Prove the following are valid arguments by giving formal proof. a. If the rain does not come, the crops are ruined and the people will starve. The crops are not ruined or the people will not starve. Therefore, the rain comes. b. If the team is late, then it cannot play the game. If the referee is here then the team can play the game. The team is late. Therefore, the referee is not here. 1.4. Set theory In this section, we study some part of set theory especially description of sets, Venn diagrams and operations of sets. Section objectives: After completing this section, students will be able to:-  Explain the concept of set.  Describe sets in different ways. 25   Identify operations on sets.  Illustrate sets using Venn diagrams. 1.4.1. The concept of a set The term set is an undefined term, just as a point and a line are undefined terms in geometry. However, the concept of a set permeates every aspect of mathematics. Set theory underlies the language and concepts of modern mathematics. The term set refers to a well-defined collection of objects that share a certain property or certain properties. The term “well-defined” here means that the set is described in such a way that one can decide whether or not a given object belongs in the set. If is a set, then the objects of the collection are called the elements or members of the set. If is an element of the set , we write. If is not an element of the set , we write. As a convention, we use capital letters to denote the names of sets and lowercase letters for elements of a set. Note that for each objects and each set , exactly one of or but not both must be true. 1.4.2. Description of sets Sets are described or characterized by one of the following four different ways. 1. Verbal Method In this method, an ordinary English statement with minimum mathematical symbolization of the property of the elements is used to describe a set. Actually, the statement could be in any language. Example 1.23: a. The set of counting numbers less than ten. b. The set of letters in the word “Addis Ababa.” c. The set of all countries in Africa. 2. Roster/Complete Listing Method If the elements of a set can all be listed, we list them all between a pair of braces without repetition separating by commas, and without concern about the order of their appearance. Such a method of describing a set is called the roster/complete listing method. Examples 1.24: a. The set of vowels in English alphabet may also be described as. b. The set of positive factors of 24 is also described as. Remark: 26  i. We agree on the convention that the order of writing the elements in the list is immaterial. As a result the sets and contain the same elements, namely and ii. The set contains just two distinct elements; namely and , hence it is the same set as We list distinct elements without repetition. Example 1.25: a. Let Elements of are and Notice that and are different objects. Here but. b. Let. The only element of is. But. c. Let Then C has four elements. The readers are invited to write down all the elements of C. 3. Partial Listing Method In many occasions, the number of elements of a set may be too large to list them all; and in other occasions there may not be an end to the list. In such cases we look for a common property of the elements and describe the set by partially listing the elements. More precisely, if the common property is simple that it can easily be identified from a list of the first few elements, then with in a pair of braces, we list these few elements followed (or preceded) by exactly three dotes and possibly by one last element. The following are such instances of describing sets by partial listing method. Example 1.26: a. The set of all counting numbers is. b. The set of non-positive integers is. c. The set of multiples of 5 is. d. The set of odd integers less than 100 is 4. Set-builder Method When all the elements satisfy a common property , we express the situation as an open proposition and describe the set using a method called the Set-builder Method as follows: We read it as “ is equal to the set of all ’s such that is true.” Here the bar and the colon “ ” mean “such that.” Notice that the letter is only a place holder and can be replaced throughout by other letters. So, for a property , the set { and are all the same set. Example 1.27: The following sets are described using the set-builder method. a.. b. c. 27  d. e. Exercise: Express each of the above by using either the complete or the partial listing method. Definition 1.10: The set which has no element is called the empty (or null) set and is denoted by or. Example 1.28: The set of such that is an empty set. Definition 1.11: A set is finite if it has limited number of elements and it is called infinite if it has unlimited number of elements. Relationships between two sets Definition 1.12: Set is said to be a subset of set (or is contained in ), denoted by , if every element of is an element of , i.e.,. It follows from the definition that set is not a subset of set if at least one element of is not an element of. i.e.,. In such cases we write or. Remarks: For any set and. Example 1.29: a. If , and , then and On the other hand, it is clear that: , and. b. If and , then since every multiple of 6 is even. However, while. Thus. c. If then and. On the other hand, since , , and. Definition 1.13: a. Sets and are said to be equal if they contain exactly the same elements. In this case, we write. That is,. b. Sets and are said to be equivalent if and only if there is a one to one correspondence among their elements. In this case, we write. Example 1.30: a. The sets are all equal. b. 28 Definition 1.14: Set is said to be a proper subset of set if every element of is also an element of , but has at least one element that is not in. In this case, we write. We also say is a proper super set of A, and write. It is clear that. Remark: Some authors do not use the symbol. Instead they use the symbol for both subset and proper subset. In this material, we prefer to use the notations commonly used in high school mathematics, and we continue using and differently, namely for subset and proper subset, respectively. Definition 1.15: Let be a set. The power set of , dented by , is the set whose elements are all subsets of. That is,. Note: If a set is finite with elements, then a. The number of subsets of is and b. The number proper subsets of is. Example 1.31: Let. As noted before, and are subset of. Moreover, and are also subsets of. Therefore,. Frequently it is necessary to limit the topic of discussion to elements of a certain fixed set and regard all sets under consideration as a subset of this fixed set. We call this set the universal set or the universe and denoted by. Exercises 1. Which of the following are sets? c. 1,2,3 d. {1,2},3 e. {{1},2},3 f. {1,{2},3} g. {1,2,a,b}. 2. Which of the following sets can be described in complete listing, partial listing and/or set-builder methods? Describe each set by at least one of the three methods. a. The set of the first 10 letters in the English alphabet. b. The set of all countries in the world. c. The set of students of Addis Ababa University in the 2018/2019 academic year. d. The set of positive multiples of 5. e. The set of all horses with six legs. 3. Write each of the following sets by listing its elements within braces. 29  c. d. e. f. g.. 4. Let be the set of positive even integers less than 15. Find the truth value of each of the following. a. b. c. d. e. f. g. h. i. 5. Find the truth value of each of the following and justify your conclusion. a.  b. c. for any set A d. , for any set A e. f. g. For any set h. 6. For each of the following set, find its power set. a. b. c. d. 7. How many subsets and proper subsets do the sets that contain exactly and elements have? 8. Is there a set A with exactly the following indicated property? a. Only one subset b. Only one proper subset 30  c. Exactly 3 proper subsets d. Exactly 4 subsets e. Exactly 6 proper subsets f. Exactly 30 subsets g. Exactly 14 proper subsets h. Exactly 15 proper subsets 9. How many elements does A contain if it has: a. 64 subsets? b. 31 proper subsets? c. No proper subset? d. 255 proper subsets? 10. Find the truth value of each of the following. a. b. c. For any set d. For any set 11. For any three sets , and , prove that: a. If and , then. b. If and , then. 1.4.3. Set Operations and Venn diagrams Given two subsets and of a universal set , new sets can be formed using and in many ways, such as taking common elements or non-common elements, and putting everything together. Such processes of forming new sets are called set operations. In this section, three most important operations, namely union, intersection and complement are discussed. Definition 1.16: The union of two sets and , denoted by , is the set of all elements that are either in or in (or in both sets). That is,. As easily seen the union operator “ ” in the theory of set is the counterpart of the logical operator “ ”. Definition 1.17: The intersection of two sets and , denoted by , is the set of all elements that are in and. That is,. 31 As suggested by definition 1.17, the intersection operator “ ” in the theory of sets is the counterpart of the logical operator “ ”. Note: - Two sets and are said to be disjoint sets if. Example 1.32: a. Let and. Then, and. b. Let = The set of positive even integers, and = The set of positive multiples of 3. Then, Definition 1.18: The difference between two sets and , denoted by , is the of all elements in and not in ; this set is also called the relative complement of with respect to. Symbolically,. Note: is sometimes denoted by. and are used interchangeably. Example 1.33: If , , then and. Note: The above example shows that, in general, are disjoint. Definition 1.19: Let be a subset of a universal set. The absolute complement (or simply complement) of , denoted by (or or , is defined to be the set of all elements of that are not in. That is, or. Notice that taking the absolute complement of is the same as finding the relative complement of with respect to the universal set. That is,. Example 1.34: a. If , and if , then. b. Let and. Then, , , 32  , , , and c. Let and. Then , , , , and. Find , ,. Which of these are equal? Theorem 1.1: For any two sets and , each of the following holds. 1.. 2.. 3.. 4.. 5.. 6.. Now we define the symmetric difference of two sets. Definition 1.20: The symmetric difference of two sets and , denoted by , is the set. Example 1.35: Let be the universal set, and. Then and. Thus. Theorem 1.2: For any three sets , and , each of the following holds. a.. ( is commutative) b.. ( is commutative) c.. ( is associative) d.. ( is associative) e.. ( is distributive over ) f.. ( is distributive over ) Let us prove property “e” formally. (definition of ) (definition of )  ( is distributive over )  ) (definition of ) 33  (definition of ) Therefore, we have. The readers are invited to prove the rest part of theorem (1.2). Venn diagrams While working with sets, it is helpful to use diagrams, called Venn diagrams, to illustrate the relationships involved. A Venn diagram is a schematic or pictorial representative of the sets involved in the discussion. Usually sets are represented as interlocking circles, each of which is enclosed in a rectangle, which represents the universal set. In some occasions, we list the elements of set inside the closed curve representing. Example 1.36: a. If and , then a Venn diagram representation of these two sets looks like the following. b. Let. A Venn diagram representation of these sets is given below. 34 Example 1.37: Let U = The set of one digits numbers A = The set of one digits even numbers B = The set of positive prime numbers less than 10 We illustrate the sets using a Venn diagram as follows. A B U 0 4 3 1 2 6 5 9 8 7 a. Illustrate by a Venn diagram A B U A  B : The shaded portion b. Illustrate A’ by a Venn diagram U A A’ : The shaded portion c. Illustrate A\B by using a Venn diagram 35  A B U A \ B: The shaded portion Now we illustrate intersections and unions of sets by Venn diagram. Cases Shaded is Shaded Only some A B A B common elements B B A A A B A B No common element A  B = Exercises 1. If , and , find. 2. Let , and { or }. Find a.. 36  b. Is ? 3. Suppose The set of one digit numbers and { is an even natural number less than or equal to 9} Describe each of the sets by complete listing method: a.. b.. c.. d. e.. f. g. 4. Suppose The set of one digit numbers and { is an even natural number less than or equal to 9} Describe each of the sets by complete listing method: h.. i.. j.. k. l.. m. n. 5. Use Venn diagram to illustrate the following statements: a.. b.. c. If , then. d.. 6. Let and. Then show that. 7. Perform each of the following operations. a. b. c. d. 8. Let { is a positive prime factor of 66} { is composite number } and. Then find each of the following. 37 9. Let and. a. , then b. , then c. , then 10. Let and. Verify each of the following. a.. b.. c. d. e. 11. Depending on question No. 10 find. a.. b.. c.. d. 12. For any two subsets and of a universal set , prove that: a.. b.. c.. d.. 13. Draw an appropriate Venn diagram to depict each of the following sets. a. U = The set of high school students in Addis Ababa. A = The set of female high school students in Addis Ababa. B = The set of high school anti-AIDS club member students in Addis Ababa. C = The set of high school Nature Club member students in Addis Ababa. b. U = The set of integers. A = The set of even integers. B = The set of odd integers. C = The set of multiples of 3. D = The set of prime numbers. 38  Chapter 2 The Real and Complex Number Systems In everyday life, knowingly or unknowingly, we are doing with numbers. Therefore, it will be nice if we get familiarized with numbers. Whatever course (which needs the concept of mathematics) we take, we face with the concept of numbers directly or indirectly. For this purpose, numbers and their basic properties will be introduced under this chapter. Objective of the Chapter At the end of this chapter, students will be able to: - check the closure property of a given set of numbers on some operations - determine the GCF and LCM of natural numbers - apply the principle of mathematical induction to prove different mathematical formulae - determine whether a given real number is rational number or not - plot complex numbers on the complex plane - convert a complex number from rectangular form to polar form and vice-versa - extract roots of complex numbers 2.1 The real number System 2.1.1 The set of natural numbers The history of numbers indicated that the first set of numbers used by the ancient human beings for counting purpose was the set of natural (counting) numbers. Definition 2.1.1 The set of natural numbers is denoted by N and is described as N =  1, 2, 3,  2.1.1.1 Operations on the set of natural numbers i) Addition (+) If two natural numbers a & b are added using the operation “+”, then the sum a+b is also a natural number. If the sum of the two natural numbers a & b is denoted by c, then we can write the operation as: c = a+b, where c is called the sum and a & b are called terms. Example: 3+8 = 11, here 11 is the sum whereas 3 & 8 are terms. ii) Multiplication (  ) If two natural numbers a & b are multiplied using the operation “  ”, then the product a  b is also a natural number. If the product of the two natural numbers a & b is denoted by c, then we can write the operation as: c = a  b, where c is called the product and a & b are called factors. Example 2.1.3: 3  4 = 12, here 12 is the product whereas 3 & 4 are factors. 39 Properties of addition and multiplication on the set of natural numbers i. For any two natural numbers a & b, the sum a+b is also a natural number. For instance in the above example, 3 and 8 are natural numbers, their sum 11 is also a natural number. In general, we say that the set of natural numbers is closed under addition. ii. For any two natural numbers a & b, a + b = b + a. Example 2.1.1: 3+8 = 8+3 = 11. In general, we say that addition is commutative on the set of natural numbers. iii. For any three natural numbers a, b & c, (a+b)+c = a +(b+c). Example 2.1.2: (3+8)+6 = 3+(8+6) = 17. In general, we say that addition is associative on the set of natural numbers. iv. For any two natural numbers a & b, the product a  b is also a natural number. For instance in the above example, 3 and 4 are natural numbers, their product 12 is also a natural number. In general, we say that the set of natural numbers is closed under multiplication. v. For any two natural numbers a & b, a  b = b  a. Example 2.1.4: 3  4 = 4  3 = 12. In general, we say that multiplication is commutative on the set of natural numbers. vi. For any three natural numbers a, b & c, (a  b)  c = a  (b  c). Example 2.1.5: (2  4)  5 = 2  (4  5) = 40. In general, we say that multiplication is associative on the set of natural numbers. vi. For any natural number a, it holds that a  1 = 1  a = a. Example 2.1.6: 6  1 = 1  6 = 6. In general, we say that multiplication has an identity element on the set of natural numbers and 1 is the identity element. vii. For any three natural numbers a, b & c, a  (b+c) = (a  b)+(a  c). Example 2.1.7: 3  (5+7) = (3  5)+ (3  7) = 36. In general, we say that multiplication is distributive over addition on the set of natural numbers. Note: Consider two numbers a and b, we say a is greater than b denoted by a  b if a – b is positive. 2.1.1.2 Order Relation in N i) Transitive property: For any three natural numbers a, b & c, a  b & b  c  a  c ii) Addition property: For any three natural numbers a, b & c, a  b  a  c  b  c iii) Multiplication property: For any three natural numbers a, b and c, a  b  ac  bc iv) Law of trichotomy For any two natural numbers a & b we have a  b or a  b or a  b. 40 2.1.1.3 Factors of a number Definition 2.2 If a, b, c  N such that ab  c , then a & b are factors (divisors) of c and c is called product (multiple) of a & b. Example 2.8: Find the factors of 15. Solution: Factors of 15 are 1, 3, 5, 15. Or we can write it as : F15   1, 3, 5, 15  Definition 2.3 A number a  N is said to be i. Even if it is divisible by 2. ii. Odd if it is not divisible by 2. iii. Prime if it has only two factors (1 and itself). iv. Composite: if it has three or more factors. Example 2.9: 2, 4, 6,... are even numbers Example 2.10: 1, 3, 5,... are odd numbers Example 2.11: 2, 3, 5,... are prime numbers Example 2.12: 4, 6, 8, 9,... are composite numbers Remark: 1 is neither prime nor composite. 2.1.1.5 Prime Factorization Definition 2.4 Prime factorization of a composite number is the product of all its prime factors. Example 2.9: a) 6  2  3 b) 30  2  3  5 c) 12  2  2  3  2 2  3 d ) 8  2  2  2  2 3 e) 180  2 2  3 2  5 Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of its prime factors. This factorization is unique except the order of the factors. 41 2.1.1.6 Greatest Common Factor (GCF) Definition 2.5 The greatest common factor (GCF) of two numbers a & b is denoted by GCF (a, b) and is the greatest number which is a factor of each of the given number. Note: If the GCF of two numbers is 1, then the numbers are called relatively prime. Example 2.10: Consider the two numbers 24 and 60. Now F24   1, 2, 3, 4, 6, 8, 12, 24  and F60   1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60  Next F24  F60   1, 2, 3, 4, 6, 12  from which 12 is the greatest. Therefore, GCF(24, 60) = 12. This method of finding the GCF of two or more numbers is usually lengthy and time consuming. Hence an alternative method (Prime factorization method) is provided as below: Step 1: Find the prime factorization of each of the natural numbers Step 2: Form the GCF of the given numbers as the product of every factor that appears in each of the prime factorization but take the least number of times it appears. Example 2.11: Consider the two numbers 24 and 60. Step1 : 24  2 3  3 60  2 2  3  5 Step 2: The factors that appear in both cases are 2 and 3, but take the numbers with the least number of times.  GCF (24, 60)  2 2  3  12 Example 2.12: Consider the three numbers 20, 80 and 450. Step1 : 20  2 2  5 80  2 4  5 450  2  3 2  5 2 Step 2: The factors that appear in all cases are 2 and 5, but take the numbers with the least number of times.  GCF (20, 80, 450)  2  5  10 2.1.1.7 Least Common Multiple (LCM) Definition 2.6 The least common multiple (LCM) of two numbers a & b is denoted by LCM (a, b) and is the least number which is a multiple of each of the given number. Example 2.13: Consider the two numbers 18 and 24. 42 Now M 18   18, 36, 54, 72, 90, 108, 126, 144,   and M 24   24, 48, 72, 96,120, 144,   Next M 18  M 24   72, 144,   from which 72 is the least. Therefore, LCM (18, 24) = 72. This method of finding the LCM of two or more numbers is usually lengthy and time consuming. Hence an alternative method (Prime factorization method) is provided as below: Step 1: Find the prime factorization of each of the natural numbers Step 2: Form the LCM of the given numbers as the product of every factor that appears in any of the prime factorization but take the highest number of times it appears. Example 2.14: Consider the two numbers 18 and 24. Step1 : 18  2 2  3 2 24  2 3  3 Step 2: The factors that appear in any case are 2 and 3, but take the numbers with the highest number of times.  LCM (18, 24)  2 3  32  72 Example 2.15: Consider the three numbers 20, 80 and 450. Step1 : 20  2 2  5 80  2 4  5 450  2  3 2  5 2 Step 2: The factors that appear in any cases are 2 , 3 and 5, but take the numbers with the highest number of times.  LCM (20, 80, 450)  2 4  32  5 2  3600 2.1.1.8 Well ordering Principle in the set of natural numbers Proposition 2.7 Every non-empty subset of the set of natural numbers has smallest (least) element. Example 2.16 A   2 , 3, 4,    N. smallest element of A  2. Note: The set of counting numbers including zero is called the set of whole numbers and is denoted by W. i.e W =  0, 1, 2, 3,  43 2.1.1.9 Principle of Mathematical Induction Mathematical induction is one of the most important techniques used to prove in mathematics. It is used to check conjectures about the outcome of processes that occur repeatedly according to definite patterns. We will introduce the technique with examples. For a given assertion involving a natural number n, if i. the assertion is true for n = 1 (usually). ii. it is true for n = k+1, whenever it is true for n = k (k  1), then the assertion is true for every natural number n. The method is used to prove different propositions involving positive integers using three steps: Step1: Prove that Tk (usually T1 ) holds true. Step 2: Assume that Tk for k = n is true. Step 3: Show that Tk is true for k = n+1. Example 2.17 Show that 1  3  5    (2n  1)  n 2. Proof: Step1. For n  1, 1  12 which is true. Step 2. Assume that it is true for n  k i.e. 1  3  5    (2k  1)  k 2. Step3. We should show that it is true for n  k  1. Claim :1  3  5    (2k  1)  (2k  1)  (k  1) 2 Now 1  3  5    (2k  1)  (2k  1)  k 2  (2k  1)  k 2  2k  1  (k  1) 2 which is the required result.  It is true for any natural number n. n (n  1) Example 2.18 Show that 1  2  3    (n) . 2 Proof: 1(1  1) Step1. For n  1, 1  which is true. 2 Step2. Assume that it is true for n  k k ( k  1) i.e. 1  2  3    (k ) . 2 Step3. We should show that it is true for n  k  1 44  ( k  1) ( k  2) Claim :1  2  3    ( k )  ( k  1) . 2 k ( k  1) Now 1  2  3    ( k )  ( k  1)   ( k  1) 2 k ( k  1)  2 ( k  1)  2 ( k  1)(k  2)  which is the required result. 2  It is true for any natural number n. Example 2.19 Show that 5n  6 n  9 n for n  2. Proof: Step 1. For n  2, 61  81 which is true Step 2. Assume that it is true for n  k. i.e. 5 k  6 k  9 k. Step3. We should show that it is true for n  k  1 Claim : 5 k 1  6 k 1  9 k 1. Now 5 k 1  6 k 1  5.5 k  6. 6 k  6.5 k  6. 6 k  6(5 k  6 k )  9 (5 k  6 k )  9(9 k )  9 k 1  5 k 1  6 k 1  9 k 1 which is the required format.  It is true for any natural number n  2. 2.1.2 The set of Integers As the knowledge and interest of human beings increased, it was important and obligatory to extend the natural number system. For instance to solve the equation x+1= 0, the set of natural numbers was not sufficient. Hence the set of integers was developed to satisfy such extended demands. Definition 2.8 The set of integers is denoted by Z and described as Z = ... ,2,  1, 0, 1, 2,   2.1.2.1 Operations on the set of integers i) Addition (+) 45 If two integers a & b are added using the operation “+”, then the sum a+b is also an integer. If the sum of the two integers a & b is denoted by c, then we can write the operation as: c = a+b, where c is called the sum and a & b are called terms. Example 2.20: 4+9 = 13, here 13 is the sum whereas 4 & 9 are terms. ii) Subtraction (  ) For any two integers a & b, the operation of subtracting b from a, denoted by a  b is defined by a  b  a  (b). This means that subtracting b from a is equivalent to adding the additive inverse of b to a. Example 2.21: 7  5  7  (5)  2 iii) Multiplication (  ) If two integers a & b are multiplied using the operation “  ”, then the product a  b is also an integer. If the product of the two integers a & b is denoted by c, then we can write the operation as: c = a  b, where c is called the product and a & b are called factors. Example 2.22: 4  7 = 28, here 28 is the product whereas 4 & 7 are factors. Properties of addition and multiplication on the set of integers i. For any two integers a & b, the sum a+b is also an integer. For instance in the above example, 4 and 9 are integers, their sum 13 is also an integer. In general, we say that the set of integers is closed under addition. ii. For any two integers a & b, a+b = b+a. Example 2.23: 4+9 = 9+4 = 13. In general, we say that addition is commutative on the set of integers. iii. For any three integers a, b & c, (a+b)+c = a+(b+c). Example 2.24: (5+9)+8 = 5+(9+8) = 22. In general, we say that addition is associative on the set of integers. iv. For any integer a, it holds that a+0 = 0+a = a. Example 2.25: 7+0 = 0+7 = 7. In general, we say that addition has an identity element on the set of integers and 0 is the identity element. v. For any integer a, it holds that a  (a)  a  a  0. Example 2.26: 4+-4 = -4+4 = 0. In general, we say that every integer a has an additive inverse denoted by  a. vi. For any two integers a & b, the product a  b is also an integer. For instance in the above example, 4 and 7 are integers, their product 28 is also an integer. In general, we say that the set of integers is closed under multiplication. 46 vii. For any two integers a & b, a  b = b  a. Example 2.27: 4  7 = 7  4 = 28. In general, we say that multiplication is commutative on the set of integers. viii. For any three integers a, b & c, (a  b)  c = a  (b  c). Example 2.28: (3  5)  4 = 3  (5  4) = 60. In general, we say that multiplication is associative on the set of integers. ix. For any integer a, it holds that a  1 = 1  a = a. Example 2.29: 5  1 = 1  5 = 5. In general, we say that multiplication has an identity element on the set of integers and 1 is the identity element. x. For any three integers a, b & c, a  (b+c) = (a  b)+(a  c). Example 2.30: 4  (5+6) = (4  5)+ (4  6) = 44. In general, we say that multiplication is distributive over addition on the set of integers. 2.1.2.2 Order Relation in Z i) Transitive property: For any three integers a, b & c, a  b & b  c  a  c ii) Addition property: For any three integers a, b & c, a  b  a  c  b  c iii) Multiplication property: For any three integers a, b and c, where c>0, a  b  ac  bc iv) Law of trichotomy: For any two integers a & b we have a  b or a  b or a  b. Exercise 2.1 1. Find an odd natural number x such that LCM (x, 40) = 1400. 2. There are between 50 and 60 number of eggs in a basket. When Loza counts by 3’s, there are 2 eggs left over. When she counts by 5’s, there are 4 left over. How many eggs are there in the basket? 3. The GCF of two numbers is 3 and their LCM is 180. If one of the numbers is 45, then find the second number. 4. Using Mathematical Induction, prove the following: a ) 6 n  1 is divisible by 5 , for n  0. b) 2 n  (n  1) ! , for n  0 c) x n  y n is divisible by x  y for odd natural number n  1. d) 2  4  6    2n  n(n  1) n (n  1)(2n  1) e) 12  2 2  3 2    n 2  6 47  n 2 (n  1) 2 f) 13  2 3  33    n 3  4 1 1 1 1 n g)     1 2 2  3 3  4 n(n  1) n  1 2.1.3 The set of rational numbers As the knowledge and interest of human beings increased with time, it was again necessary to extend the set of integers. For instance to solve the equation 2x+1= 0, the set of integers was not sufficient. Hence the set of rational numbers was developed to satisfy such extended needs. Definition 2.9 a Any number that can be expressed in the form , where a and b are integers and b  0 , is called a b rational number. The set of rational numbers denoted by Q is described by a  Q =  : a and b are integers and b  0 . b  Notes: a i. From the expression , a is called numerator and b is called denominator. b a ii. A rational number is said to be in lowest form if GCF (a, b) = 1. b 2.1.3.1 Operations on the set of rational numbers i) Addition (+) If two rational numbers a / b and c / d are added using the operation “+”, then the sum defined as a c ad  bc   is also a rational number. b d bd 1 3 11 Example 2.31:   2 5 10 ii) Subtraction (  ) For any two rational numbers a / b & c / d , the operation of subtracting c / d from a / b , denoted by a / b - c / d is defined by a / b - c / d = a / b +(- c / d ). 1 3 1 Example 2.32:   2 5 10 48 iii) Multiplication (  ) If two rational numbers a / b and c / d are multiplied using the operation “  ”, then the product a c ac defined as   is also a rational number. b d bd 1 3 3 Example 2.33:   2 5 10 iv) Division (  ) For any two rational numbers a / b & c / d , dividing a / b by c / d is defined by a c a d    , c  0. b d b c 1 3 1 5 5 Example 2.34:     2 5 2 3 6 Properties of addition and multiplication on the set of rational numbers Let a / b , c / d and e / f be three rational numbers, then i. The set of rational numbers is closed under addition and multiplication. ii. Addition and multiplication are both commutative on the set of rational numbers. iii. Addition and multiplication are both associative on the set of rational numbers. iv. 0 is the additive identity i.e., a / b + 0 = 0+ a / b = a / b. v. Every rational number has an additive inverse. i.e., a / b + ( a / b) =  a / b + a / b = 0. vi. 1 is the multiplicative identity i.e., a / b  1 = 1  a / b = a / b. vii. Every non-zero rational number has a multiplicative inverse. i.e., a / b  b / a = b / a  a / b = 1. 2.1.3.2 Order Relation in Q i) Transitive property For any three rational numbers a / b , c / d & e / f a / b  c / d & c / d  e / f  a / b  e / f. ii) Addition property For any three rational numbers a / b , c / d & e / f a / b  c / d  a / b  e / f  c / d  e / f. iii) Multiplication property For any three rational numbers a / b , c / d , e / f and e / f  0 49 a / b  c / d  (a / b)(e / f )  (c / d ) (e / f ). iv) Law of trichotomy For any two rational numbers a / b & c / d we have a / b  c / d or a / b  c / d or a / b  c / d. 2.1.3.3 Decimal representation of rational numbers a A rational number can be written in decimal form using long division. b 2.1.3.3.1 Terminating decimals 25 Example 2.35: Express the fraction number in decimal form. 4 25 Solution:  6.25 4 2.1.3.3.2 Non-terminating periodic decimals 25 Example 2.36: Express the fraction number in decimal form. 3 25 Solution:  8.333 3 Now we will see how to convert decimal numbers in to their fraction forms. In earlier mathematics topics, we have seen that multiplying a decimal by 10 pushes the decimal point to the right by one position and in general, multiplying a decimal by 10n pushes the decimal point to the right by n positions. We will use this fact for the succeeding topics. 2.1.3.4 Fraction form of decimal numbers A rational number which is written in decimal form can be converted to a fraction a form as in lowest (simplified) form, where a and b are relatively prime. b 2.1.3.4.1 Terminating decimals Consider any terminating decimal number d. Suppose d terminates n digits after the decimal point. d can be converted to its fraction form as below: 1 10 n d  d  1  d   d  ( n ). 1 10 Example 2.37: Convert the terminating decimal 3.47 to fraction form. 102 347 Solution: 3.47  3.47  . 102 100 50 2.1.3.4.2 Non-terminating periodic decimals Consider any non-terminating periodic decimal number d. Suppose d has k non-terminating digits and p terminating digits after the decimal point. d can be converted to its fraction form as below: 1 10 k  p  10 k d  d 1  d   d  ( k  p ). 1 10  10 k Example 2.38: Convert the non-terminating periodic decimal 42.538 to fraction form. Solution: k = 1, p = 2. 1 10k  p  10k 103  10 42538. 38  425. 38 42113  d  d 1  d   d  ( k  p )  42.538  ( ) . 1 10  10 k 10  10 3 1000  10 990 Note: From the above two cases, we can conclude that both terminating decimals and non- terminating periodic decimals are rational numbers. (Why? Justify). 2.1.3.5 Non-terminating and non-periodic decimals Some decimal numbers are neither terminating nor non-terminating periodic. Such types of numbers are called irrational numbers. Example 2.39: 62.757757775…. Example 2.40: Show that 2 is an irrational number. Proof: Suppose 2 is a rational number a  2  , where GCF (a, b)  1 b a2 2 2 b  a  2b 2.........(*) 2  a 2 is even  a is even  a  2n..............(**) Putting this in (*) we get :  4 n 2  2b 2  b 2  2n 2  b 2 is even  b is even  b  2m............(* * *) 51 From (**) and (***) we get a contradiction that GCF (a, b) = 1 which implies that 2 is not a rational number. Therefore, 2 is an irrational number. 2.1.4 The set of real numbers Definition 2.10 A number is called a real number if and only if it is either a rational number or an irrational number. The set of real numbers denoted by  can be described as the union of the set of rational and irrational numbers. i.e  = {x : x is a rational number or an irrational number}. There is a 1-1 correspondence between the set of real numbers and the number line (For each point in the number line, there is a corresponding real number and vice-versa). 2.1.4.1 Operations on the set of real numbers i) Addition (+) If two real numbers are added using the operation “+”, then the sum is also a real number. ii) Subtraction (  ) For any two real numbers a & b , the operation of subtracting b from a , denoted by a  b is defined by a  b = a +(  b ). iii) Multiplication (  ) If two real numbers a and b are multiplied using the operation “  ”, then the product defined as a  b  ab is also a real number. iv) Division (  ) 1 For any two real numbers a & b , dividing a by b is defined by a  b  a  , b  0. b Properties of addition and multiplication on the set of real numbers Let a , b and c be three real numbers, then i. The set of real numbers is closed under addition and multiplication. ii. Addition and multiplication are commutative on the set of real numbers. iii. Addition and multiplication are associative on the set of real numbers. iv. 0 is the additive identity i.e., a + 0 = 0+ a = a. v. Every real number has an additive inverse. 52  i.e., a + (  a ) =  a + a = 0. vi. 1 is the multiplicative identity i.e., a  1 = 1  a = a. vii. Every non-zero real number has a multiplicative inverse. i.e., a  1/ a = 1/ a  a = 1. 2.1.4.2 The real number and the number line One of the most important properties of the real number is that it can be represented graphically by points on a straight line. The point 0 is termed as the origin. Points to the right of 0 are called positive real numbers and points to the left of 0 are called negative real numbers. Each point on the number line corresponds a unique real number and vice-versa. Geometrically we say a is greater than b if a is located to the right of b on the number line. 2.1.4.3 Order Relation in R i) Transitive property: For any three real numbers a , b & c , a  b & b  c  a  c. ii) Addition property: For any three real numbers a , b & c , a  b  a  c  b  c. iii) Multiplication property: For any three real numbers a , b , c and c  0 , we have a  b  ac  bc. iv) Law of trichotomy: For any two real numbers a & b we have a  b or a  b or a  b. Summary of the real number system 53 2.1.4.4 Intervals Let a and b be two real numbers such that a  b, then the intervals which are subsets of R with end points a and b are denoted and defined as below: i. (a, b)   x : a  x  b open interval from a to b. ii. [a, b]   x : a  x  b closed interval from a to b. iii. (a, b]   x : a  x  b open-closed interval from a to b. iv. [a, b)   x : a  x  b closed-open interval from a to b. 2.1.4.5 Upper bounds and lower bounds Definition 2.11 Let A be non  empty and A  . i. A point a  R is said to be an upper bound of A iff x  a for all x  A. ii. An upper bound of A is said to be least upper bound (lub) iff it is the least of all upper bounds. iii. A point a  R is said to be lower bound of A iff x  a for all x  A. ii. A lower bound of A is said to be greatest lower bound (glb) iff it is the greatest of all lower bounds. Example 2.41 Consider the set A  2, 5  . i ) lower bounds are ,  9,  3, 0, 1 , 1, 2 2 Here the greatest element is 2.  glb  2 25 ii) upper bounds are 5, 6, , 20, 99,1000 3 Here the least element is 5.  lub  5. Example 2.42: Consider the set A =   for n  N. 1 n Solution: A   1, , ,  1 1  2 3  i ) lower bounds are ,  3,  2, 0 Here the greatest element is 0. Thus, glb  0 9 ii) upper bounds are 1, 3, , 50,  2 Here the least element is 1. Thus, lub  1. 54 Based on the above definitions, we can define the completeness property of real numbers as below. 2.1.4.4 Completeness property of real number (R) Completeness property of real numbers states that: Every non-empty subset of  that has lower bounds has glb and every non-empty subset of  that has upper bounds has a lub. Exercise 2.2 1. Express each of the following rational numbers as decimal: 4 3 11 2 2