GE 112: Mathematics In The Modern World (MODULE 2) PDF

GE 112: MATHEMATICS IN THE MODERN WORLD (MODULE 2: MATHEMATICAL LANGUAGE AND SYMBOLS) Jan Carl M. Vertudes Part-time Lecturer, Mathematics and Statistics Deparment College of Arts and Sciences University of Southeastern Philippi...

GE 112: MATHEMATICS IN THE MODERN WORLD (MODULE 2: MATHEMATICAL LANGUAGE AND SYMBOLS) Jan Carl M. Vertudes Part-time Lecturer, Mathematics and Statistics Deparment College of Arts and Sciences University of Southeastern Philippines - Main Campus GE 112: MATHEMATICS IN THE MODERN WORLD In this Module, we will discuss the following topics: 1 Mathematical Language 2 Elementary Logic 3 Sets 4 Functions and Relations GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language 2.1. Mathematical Language GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Mathematical Language The Importance of Language Although ideas may be simple, there is no access to the ideas without a knowledge of the language in which the ideas are expressed. For example, people frequently have trouble understanding mathematical ideas: not necessarily because the ideas are difficult, but because they are being presented in a foreign language – the language of mathematics. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language What is language in mathematics? GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language What is language in mathematics? Language in mathematics is a system of com- munication about objects like numbers, variables, sets, operations, functions and equations. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language What is language in mathematics? Language in mathematics is a system of com- munication about objects like numbers, variables, sets, operations, functions and equations. Mathematics has its own vocabulary, gram- mar, syntax, etc. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Characteristics of Mathematical Language The language of mathematics makes it easy to ex- press the kinds of thoughts that mathematicians like to express. It is: GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Characteristics of Mathematical Language The language of mathematics makes it easy to ex- press the kinds of thoughts that mathematicians like to express. It is: 1. Precise (able to make very fine distinctions) GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Characteristics of Mathematical Language The language of mathematics makes it easy to ex- press the kinds of thoughts that mathematicians like to express. It is: 1. Precise (able to make very fine distinctions) 2. Concise (able to say things briefly). GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Characteristics of Mathematical Language The language of mathematics makes it easy to ex- press the kinds of thoughts that mathematicians like to express. It is: 1. Precise (able to make very fine distinctions) 2. Concise (able to say things briefly). 3. Powerful (able to express complex thoughts with relative ease). GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Mathematical Symbols and Notations Symbols for operations: +, −, ×, ÷ GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Mathematical Symbols and Notations Symbols for operations: +, −, ×, ÷ Symbol that represents values: a, b, c, x, y, z, etc. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Mathematical Symbols and Notations Symbols for operations: +, −, ×, ÷ Symbol that represents values: a, b, c, x, y, z, etc. Other Symbols: N - set of natural number {1, 2, 3, 4,...} W - set of whole number {0, 1, 2, 3, 4,...} Z - set of integers {... , −2, −1, 0, 1, 2,...} GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language R - set of real numbers (−∞, ∞) Q - set of rational number { ab : a, b ∈ Z ∧ b ̸= 0} QC - set of irrational number {x|x is nonrepeating and nonterminating decimal} C - set of complex number {x + iy : x, y ∈ R, i2 = −1} ∧ - logical “and” ∨ - logical “or” ∈ - an element ∈ / - not an element GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language → - if - then ↔ - if and only if ∀ - for any, for all ∃ - there exist, there is ∴ - therefore - summation P ∩ - intersection ∪ - union ∅ - empty set, null set ⊆ - subset ⊂ - proper subset GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Expression vs. Sentence Nouns and Sentences in the English Language In English, nouns are used to name things we want to talk about (like people, places, and things); whereas sentences are used to state complete thoughts. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.1 For example, consider the sentence Carol loves mathematics. Here, ‘Carol’ and ‘mathematics” are nouns; ‘loves’ is a verb. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Expression versus Sentences in Mathematics The mathematical analogue of a ‘noun’ will be called an expression. Thus, an expression is a name given to a mathematical object of interest. The mathematical analogue of a ‘sentence’ will also be called a sentence. A mathematical sentence, just as an English sentence, must state a complete thought. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language The table below summarizes the analogy between the English and Mathematics (don’t worry for the mo- ment about the truth of sentences). GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.2 For example, the expressions 5 2+3 10 ÷ 2 (6 − 2) + 1 1+1+1+1+1 all look different, but are all just different names for the same number. English has the same concept (see synonyms). GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Sentences have Verbs Just as English sentences have verbs, so do mathematical sentences. In the mathematical sentence ‘3 + 4 = 7’, the verb is ‘=’. Indeed, the equal sign is one of the most popular mathe- matical verbs. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Truth of Sentences Sentences can be true or false. The notion of truth (i.e. the property of being true or false) is of fundamental importance in the mathematical language. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language 1. Identify the verbs in the following sentences: a. The capital of Philippines is Manila. b. The capital of Japan is Nagasaki. c. 3+4=7 d. 3+4=8 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language 1. Identify the verbs in the following sentences: a. The capital of Philippines is Manila. b. The capital of Japan is Nagasaki. c. 3+4=7 d. 3+4=8 2. True or False: a. The capital of Philippines is Manila. b. The capital of Japan is Nagasaki. c. 3+4=7 d. 3+4=8 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 2. 2 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 2. 2 3. The word ‘cat’ begins with the letter ‘k’ GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 2. 2 3. The word ‘cat’ begins with the letter ‘k’ 4. 1 + 2 = 4 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 2. 2 3. The word ‘cat’ begins with the letter ‘k’ 4. 1 + 2 = 4 5. 5 − 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 6. The cat is black 2. 2 3. The word ‘cat’ begins with the letter ‘k’ 4. 1 + 2 = 4 5. 5 − 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 6. The cat is black 2. 2 7. x 3. The word ‘cat’ begins with the letter ‘k’ 4. 1 + 2 = 4 5. 5 − 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 6. The cat is black 2. 2 7. x 3. The word ‘cat’ begins 8. x − 1 = 0 with the letter ‘k’ 4. 1 + 2 = 4 5. 5 − 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 6. The cat is black 2. 2 7. x 3. The word ‘cat’ begins 8. x−1=0 with the letter ‘k’ 9. t+3=3+t 4. 1 + 2 = 4 5. 5 − 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.3 If possible, classify the entries in the list below as an English noun, mathematical expression, English sen- tence, or a mathematical sentence. 1. cat 6. The cat is black 2. 2 7. x 3. The word ‘cat’ begins 8. x−1=0 with the letter ‘k’ 9. t+3=3+t 4. 1 + 2 = 4 10. 1·x=x 5. 5 − 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.4 For the following sentences, identify which are true or false. Are there possibilities other than true and false? 1. The word ‘cat’ begins with letter ‘k’. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.4 For the following sentences, identify which are true or false. Are there possibilities other than true and false? 1. The word ‘cat’ begins with letter ‘k’. 2. 1 + 2 = 4 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.4 For the following sentences, identify which are true or false. Are there possibilities other than true and false? 1. The word ‘cat’ begins with letter ‘k’. 2. 1 + 2 = 4 3. The cat has four legs. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.4 For the following sentences, identify which are true or false. Are there possibilities other than true and false? 1. The word ‘cat’ begins with letter ‘k’. 2. 1+2=4 3. The cat has four legs. 4. x − 1 = −1 + x GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.4 For the following sentences, identify which are true or false. Are there possibilities other than true and false? 1. The word ‘cat’ begins with letter ‘k’. 2. 1+2=4 3. The cat has four legs. 4. x − 1 = −1 + x 5. t+3=3+t GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Example 1.4 For the following sentences, identify which are true or false. Are there possibilities other than true and false? 1. The word ‘cat’ begins with letter ‘k’. 2. 1+2=4 3. The cat has four legs. 4. x − 1 = −1 + x 5. t+3=3+t 6. 1·x=x GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Expression vs. Sentences Expression is a name given to a mathematical object of interest. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Expression vs. Sentences Expression is a name given to a mathematical object of interest. Example: 5, 2 + 3, 10 ÷ 2, (6 − 2 + 1), 2 + 2 + 2 + 2 + 2 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Expression vs. Sentences Expression is a name given to a mathematical object of interest. Example: 5, 2 + 3, 10 ÷ 2, (6 − 2 + 1), 2 + 2 + 2 + 2 + 2 Mathematical sentence, just an English sentence, must state a complete thought. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Expression vs. Sentences Expression is a name given to a mathematical object of interest. Example: 5, 2 + 3, 10 ÷ 2, (6 − 2 + 1), 2 + 2 + 2 + 2 + 2 Mathematical sentence, just an English sentence, must state a complete thought. Example: 2+3 = 5, 10÷2 = 5, (6−2)+3 = 7, 2+2+1+2+3+4 = 14 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Difficulties The word “is” could mean equality, inequality or membership in a set. Different use of a number (cardinal, ordinal, nomi- nal, ratio). Mathematical objects may be represented in many ways such as sets and functions. The words “and” and “or” mean differently in math- ematics from its English use. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from English Language to Math Language 1 The sum of two numbers is 10. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from English Language to Math Language 1 The sum of two numbers is 10. 2 The square root of 9 added to x is 5. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from English Language to Math Language 1 The sum of two numbers is 10. 2 The square root of 9 added to x is 5. 3 Two added to a number subtracted from 10 equals 3. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from English Language to Math Language 1 The sum of two numbers is 10. 2 The square root of 9 added to x is 5. 3 Two added to a number subtracted from 10 equals 3. 4 The set of integers is a subsets of the set of real numbers. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from English Language to Math Language 1 The sum of two numbers is 10. 2 The square root of 9 added to x is 5. 3 Two added to a number subtracted from 10 equals 3. 4 The set of integers is a subsets of the set of real numbers. 5 The sum of two consecutive odd number is eight. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from English Language to Math Language 1 The sum of two numbers is 10. 2 The square root of 9 added to x is 5. 3 Two added to a number subtracted from 10 equals 3. 4 The set of integers is a subsets of the set of real numbers. 5 The sum of two consecutive odd number is eight. 6 x belongs to the intersection of sets A and B. GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from Math Language to En- glish Language 1 20 ÷ (2 + 8) GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from Math Language to En- glish Language 1 20 ÷ (2 + 8) 2 (x − 5) + 6x GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from Math Language to En- glish Language 1 20 ÷ (2 + 8) 2 (x − 5) + 6x 3 3(x − 2) = x + 3 GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from Math Language to En- glish Language 1 20 ÷ (2 + 8) 2 (x − 5) + 6x 3 3(x − 2) = x + 3 4 x∈R⊆C GE 112: MATHEMATICS IN THE MODERN WORLD Mathematical Language Translate from Math Language to En- glish Language 1 20 ÷ (2 + 8) 2 (x − 5) + 6x 3 3(x − 2) = x + 3 4 x∈R⊆C 5 ∀x, y ∈ R, (x2 + y 2 ) = x2 + 2xy + y 2 GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic 2.2. Elementary Logic GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Logic What is logic? GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Logic Logic is systematic way of thinking that allows us to deduce new information from old information and to parse the meaning of sentences. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.1 A statement is a declarative sentence that is either true or false, but not both true and false. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.1 A statement is a declarative sentence that is either true or false, but not both true and false. Example 2.1 Determine whether each sentence is a statement. 1 Davao City is one of the cities in Mindanao. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.1 A statement is a declarative sentence that is either true or false, but not both true and false. Example 2.1 Determine whether each sentence is a statement. 1 Davao City is one of the cities in Mindanao. 2 How are you? GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.1 A statement is a declarative sentence that is either true or false, but not both true and false. Example 2.1 Determine whether each sentence is a statement. 1 Davao City is one of the cities in Mindanao. 2 How are you? 3 92 + 2 is a prime. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.1 A statement is a declarative sentence that is either true or false, but not both true and false. Example 2.1 Determine whether each sentence is a statement. 1 Davao City is one of the cities in Mindanao. 2 How are you? 3 92 + 2 is a prime. 4 x + 1 = 5. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.2 Determine whether or not the following are statement. In this case of a statement, say if it is true or false, if possible. 1 Every real number is an even integer. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.2 Determine whether or not the following are statement. In this case of a statement, say if it is true or false, if possible. 1 Every real number is an even integer. 2 If x and y are real numbers and 5x = 5y, then x = y. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.2 Determine whether or not the following are statement. In this case of a statement, say if it is true or false, if possible. 1 Every real number is an even integer. 2 If x and y are real numbers and 5x = 5y, then x = y. 3 Set Z and N are infinite. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.2 Determine whether or not the following are statement. In this case of a statement, say if it is true or false, if possible. 1 Every real number is an even integer. 2 If x and y are real numbers and 5x = 5y, then x = y. 3 Set Z and N are infinite. 4 The integer x is a multiple of 7. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.2 Determine whether or not the following are statement. In this case of a statement, say if it is true or false, if possible. 1 Every real number is an even integer. 2 If x and y are real numbers and 5x = 5y, then x = y. 3 Set Z and N are infinite. 4 The integer x is a multiple of 7. 5 Either x is a multiple of 7, or it is not. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.2 A simple statement is a statement that conveys a single idea. A compound statement is a statement that con- veys two or more ideas. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.2 A simple statement is a statement that conveys a single idea. A compound statement is a statement that con- veys two or more ideas. Example 2.3 The following are simple statements. 1 Today is Friday. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.2 A simple statement is a statement that conveys a single idea. A compound statement is a statement that con- veys two or more ideas. Example 2.3 The following are simple statements. 1 Today is Friday. 2 It is raining. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.2 A simple statement is a statement that conveys a single idea. A compound statement is a statement that con- veys two or more ideas. Example 2.3 The following are simple statements. 1 Today is Friday. 2 It is raining. 3 I am going to a movie. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.2 A simple statement is a statement that conveys a single idea. A compound statement is a statement that con- veys two or more ideas. Example 2.3 The following are simple statements. 1 Today is Friday. 2 It is raining. 3 I am going to a movie. 4 I am not going to the basketball game. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.4 The following are compound statemets: 1 Today is Friday and it is raining. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.4 The following are compound statemets: 1 Today is Friday and it is raining. 2 It is not raining and I am going to a movie. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.4 The following are compound statemets: 1 Today is Friday and it is raining. 2 It is not raining and I am going to a movie. 3 I am going to a basketball game or I am going to a movie. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Example 2.4 The following are compound statemets: 1 Today is Friday and it is raining. 2 It is not raining and I am going to a movie. 3 I am going to a basketball game or I am going to a movie. 4 If it is raining, then I am not going to the basketball game. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Definition 2.3 The truth value of a simple statement is either true (T ) or false (F ). On the other hand, the truth value of a compound statement depends on the truth value of ots simple statements and its connectives. Likewise, a truth table is a table that showsthe truth value of a compound statement for all possible truth values of its simple statements. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statement and Quantifiers Note 2.3.1 George Boole (a founder of Boolean algebra which have an applications in the areas of computer programming and the design of electronic circuits) has used lowercase p, q, r, and s to represent simple statements and the symbols ∧, ∨, →, and ↔ to represent connectives. See Table 1 in the next slide. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Example 2.5 Let us consider the simple and compound statements in Example 2.3 and Example 2.4, respectively and let us use the symbols in- troduced by George Boole. p: Today is Friday q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Example 2.5 Let us consider the simple and compound statements in Example 2.3 and Example 2.4, respectively and let us use the symbols in- troduced by George Boole. p: Today is Friday q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. 1 Today is Friday and it is raining. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Example 2.5 Let us consider the simple and compound statements in Example 2.3 and Example 2.4, respectively and let us use the symbols in- troduced by George Boole. p: Today is Friday q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. 1 Today is Friday and it is raining. 2 It is not raining and I am going to a movie. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Example 2.5 Let us consider the simple and compound statements in Example 2.3 and Example 2.4, respectively and let us use the symbols in- troduced by George Boole. p: Today is Friday q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. 1 Today is Friday and it is raining. 2 It is not raining and I am going to a movie. 3 I am going to a basketball game or I am going to a movie. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Example 2.5 Let us consider the simple and compound statements in Example 2.3 and Example 2.4, respectively and let us use the symbols in- troduced by George Boole. p: Today is Friday q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. 1 Today is Friday and it is raining. 2 It is not raining and I am going to a movie. 3 I am going to a basketball game or I am going to a movie. 4 If it is raining, then I am not going to the basketball game. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Definition 2.4 Given a statement p we define the statement ∼ p (not p) to be false when p is true, and true when p is false. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Definition 2.4 Given a statement p we define the statement ∼ p (not p) to be false when p is true, and true when p is false. Example 2.6 Suppose p: I like Mathematics in the Modern World. Thus, ∼ p: I do not like Mathematics in the Modern World. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Truth Table GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Quantifiers and Negation How to write the negation of some quantified statements? GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Quantifiers and Negation How to write the negation of some quantified statements? Quantified Statements and Their Negations Displayed in a Compact Format GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Quantifiers and Negation Example 2.7 Write the negation of each of the following statements. 1 Some airpots are open GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Quantifiers and Negation Example 2.7 Write the negation of each of the following statements. 1 Some airpots are open 2 All movies are worth the price of admission. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Quantifiers and Negation Example 2.7 Write the negation of each of the following statements. 1 Some airpots are open 2 All movies are worth the price of admission. 3 No odd numbers are divisible by 2. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Definition 2.5 Given two statements p and q, we define the statement p ∧ q (p and q) to be true precisely when both p and q are true. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Definition 2.5 Given two statements p and q, we define the statement p ∧ q (p and q) to be true precisely when both p and q are true. Example 2.8 Suppose p: I like Mathematics in the Modern World. q: I like the degree program that I am taking. Thus, p ∧ q: I like Mathematics in the Modern World and the degree program that I am taking. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Truth Table GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Definition 2.6 Given two statements p and q, we define the statement p ∨ q (p or q) to be true precisely when at least onE of p and q is true. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Definition 2.6 Given two statements p and q, we define the statement p ∨ q (p or q) to be true precisely when at least onE of p and q is true. Example 2.9 Suppose p: I like Mathematics in the Modern World. q: I like the degree program that I am taking. Thus, p ∨ q: I like Mathematics in the Modern World or the degree program that I am taking. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Statements and Quantifiers Truth Table GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Compound Statements and Grouping Symbols If a compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together. Table 5 illustrates the use of parentheses to indicate groupings for some statements in symbolic form. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Compound Statements and Grouping Symbols If a compound statement is written as an English sentence, then a comma is used to indicate which simple statements are grouped together. Statements on the same side of atext comma are grouped together. See Table 6. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Example 2.10 Construct truth tables for the statements (p ∧ q) ∨ (∼ p∧ ∼ q) GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Example 2.11 Construct truth tables for the statements (∼ p ∧ q) ∨ (p ∧ r) GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Example 2.12 Determine the symbolic form of the compound statement “ Either Rπ d(2x ) x−1 Rπ −π sin(x) dx ̸= 0 and dx = x2 or −π sin(x) dx = 0 and ln(6) = ln(3) ln(2). GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Equivalent Statements and Tautologies Definition 2.7 Two statement p and q are said to be logically equivalent if they have the same truth values. If p and q are logically equivalent, we write p ⇐⇒ q. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Example 2.13 Show that the statement forms ∼ (p ∨ q) and ∼ p ∧ ∼ q are logically equivalent. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Equivalent Statements and Tautologies Definition 2.8 A statement whose truth values are all true is called tautology. The negation of a tautology is called contradiction. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Example 2.14 Show that ∼ (p ∧ q) ∨ (p ∨ q) is a tautology. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Example 2.15 Show that [(p ∨ q) ∧ (p ∨ ∼ q)] ∧ ∼ p is a contradition. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Definition 2.9 Given the statement p and q, we define: a. The statement p implies q, denoted by p → q, also read “if p, then q” is true except in the case where p is true and q is false. Such a statement is called a conditional, the component statements p and q are called the premise and conclusion, respectively. b. The statement p if and only if q, denoted by p ↔ q, also written as “p iff q”, is true precisely in this case where p and q are both true or p and q are both false. Such a statement is called a biconditional. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Truth Table for Conditional GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Truth Table for Biconditional GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Example 2.16 Show that (p → q) ∧ (q → p) is logically equivalent to p ↔ q, that is, (p → q) ∧ (q → p) ⇐⇒ (p ↔ q) GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Common Forms of p → q GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Statements related to the Conditional Statement The converse of p → q is q → p. The inverse of p → q is ∼ p →∼ q. The contrapositive of p → q is ∼ q →∼ p. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Statements related to the Conditional Statement The converse of p → q is q → p. The inverse of p → q is ∼ p →∼ q. The contrapositive of p → q is ∼ q →∼ p. Example 2.17 Write the converse, inverse and contrapositive of the statement If I get the job, then I will rent the apartment. GE 112: MATHEMATICS IN THE MODERN WORLD Elementary Logic Conditional and Biconditional Truth Table for Conditional and Related State- ment GE 112: MATHEMATICS IN THE MODERN WORLD Sets 2.3. Sets (Basic Concepts and Set Operations) GE 112: MATHEMATICS IN THE MODERN WORLD Sets Sets Sets is a well-defined, unordered and distinct collec- tion of objects. The objects that belong in a set are the elements, or members, of the set. A, B, C, D,... (capital letters) - denote a set. x, y, z,... (small letters) - denote as elements. x ∈ A - an element x is in a set A. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Common Methods of Describing a Set Roster - listing all the elements of a set inside a pair of braces,. Commas are used to separate the elements. Rule - describing the elements of a set. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Rule vs. Roster Methods GE 112: MATHEMATICS IN THE MODERN WORLD Sets Set-builders Notation Another method of representing a set is set-builders notation. Set-builder notation is especially useful when describing infinite sets. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.1 Use set-builder notation to write the following sets. 1 The set of integers greater than −3. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.1 Use set-builder notation to write the following sets. 1 The set of integers greater than −3. 2 The set of whole numbers less than 1,000. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Cardinality and Finite Sets A set is finite if the number of elements in the set is a whole number. The cardinal number of a finite set is the number of elements in the set. The cardinal number of a finite set A is denoted by the notation n(A). GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.2 Find the cardinality of each of the following sets. a. J = {2, 5} b. S = {3, 4, 5, 6, 7,... , 31} c. T = {3, 3, 7, 51} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Equal Sets Set A is equal to set B, denoted by A = B, if and only if A and B have exactly the same elements. Equivalent Sets Set A is equivalent to set B, denoted by A ∼ B, if and only if A and B have exactly the same number of elements. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.3 State whether each of the following pairs of sets are equal, equivalent, both, or neither. a. {a, e, i, o, u}, {3, 7, 11, 15, 19} b. {4, −2, 7}, {3, 4, 7, 9} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Empty Set or Null Set The empty set, or null set, is the set that contains no elements. The symbol ∅ or {} is used to represent the empty set. Universal Sets The set of all elements being considered is called a universal set. GE 112: MATHEMATICS IN THE MODERN WORLD Sets A Subset of a Set Set A is a subset of set B, denoted by A ⊆ B if and only if every element of A is also an element of B. Set A is a proper subset of set B, denoted by A ⊂ B, if every element of A is an element of B and A ̸= B. GE 112: MATHEMATICS IN THE MODERN WORLD Sets A Subset of a Set Set A is a subset of set B, denoted by A ⊆ B if and only if every element of A is also an element of B. Set A is a proper subset of set B, denoted by A ⊂ B, if every element of A is an element of B and A ̸= B. The Number of a Subset in a Set The set with n elements has 2n subsets. GE 112: MATHEMATICS IN THE MODERN WORLD Sets A Subset of a Set Set A is a subset of set B, denoted by A ⊆ B if and only if every element of A is also an element of B. Set A is a proper subset of set B, denoted by A ⊂ B, if every element of A is an element of B and A ̸= B. The Number of a Subset in a Set The set with n elements has 2n subsets. Subset Relationship 1 A ⊆ A, for any set A. 2 ∅ ⊆ A, for any set A. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.4 Determine whether each statement is true or false. 1 {5, 10, 15, 20} ⊆ {0, 5, 10, 15, 20, 25, 30} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.4 Determine whether each statement is true or false. 1 {5, 10, 15, 20} ⊆ {0, 5, 10, 15, 20, 25, 30} 2 W⊆N GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.4 Determine whether each statement is true or false. 1 {5, 10, 15, 20} ⊆ {0, 5, 10, 15, 20, 25, 30} 2 W⊆N 3 {2, 4, 6} ⊆ {2, 4, 6} 4 ∅ ⊆ {1, 2, 3} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Venn Diagram The English logician John Venn (1834-1923) devel- oped diagrams, which we now refer to as Venn dia- grams, that can be used to illustrate sets and relation- ships between sets. In a Venn diagram, the universal set is represented by a rectangular region and subsets of the universal set are generally represented by oval or circular regions drawn inside the rectangle. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Venn Diagram GE 112: MATHEMATICS IN THE MODERN WORLD Sets Complement of a Set The complement of a set A, denoted by A′ or Ac , is the set of all elements of the universal set U that are not elements of A. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Complement of a Set The complement of a set A, denoted by A′ or Ac , is the set of all elements of the universal set U that are not elements of A. Example 3.5 Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and S = {2, 4, 6, 7}. Then the complement of set S is S ′ = {1, 3, 5, 8, 9, 10}. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Intersection of Sets The intersection of sets A and B , denoted by A∩B, is the set of elements common to both A and B. In symbolic form, A ∩ B = {x|x ∈ A and x ∈ B} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Intersection of Sets The intersection of sets A and B , denoted by A∩B, is the set of elements common to both A and B. In symbolic form, A ∩ B = {x|x ∈ A and x ∈ B} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.6 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Find 1 A∩B GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.6 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Find 1 A∩B 2 A∩C GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.6 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Find 1 A∩B 2 A∩C 3 B∩C GE 112: MATHEMATICS IN THE MODERN WORLD Sets Disjoint Sets Two sets are disjoint if their intersection is the empty set. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Union of Sets The union of sets A and B , denoted by A ∪ B, is the set of that contains all the elementsthat belong to A or to B or to both. In symbolic form, A ∪ B = {x|x ∈ A or x ∈ B} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Union of Sets The union of sets A and B , denoted by A ∪ B, is the set of that contains all the elementsthat belong to A or to B or to both. In symbolic form, A ∪ B = {x|x ∈ A or x ∈ B} GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.7 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Find 1 A∪B GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.7 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Find 1 A∪B 2 A∪C GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.7 Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Find 1 A∪B 2 A∪C 3 B∪C GE 112: MATHEMATICS IN THE MODERN WORLD Sets GE 112: MATHEMATICS IN THE MODERN WORLD Sets GE 112: MATHEMATICS IN THE MODERN WORLD Sets The expression (A ∪ B)′ and A′ ∩ B ′ are both rep- resented by region iv. Thus, (A ∪ B)′ = (A′ ∩ B ′ ). GE 112: MATHEMATICS IN THE MODERN WORLD Sets The expression (A ∪ B)′ and A′ ∩ B ′ are both rep- resented by region iv. Thus, (A ∪ B)′ = (A′ ∩ B ′ ). De Morgan’s Laws For all sets A and B, (A ∪ B)′ = (A′ ∩ B ′ ) and (A ∩ B)′ = (A′ ∪ B ′ ) GE 112: MATHEMATICS IN THE MODERN WORLD Sets Venn diagrams can be used to verify each of the following properties. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.8 Let U = {1, 2, 3,... , 15}, A = {1, 3, 5, 7, 9}, B = {6, 7, 8, 9, 10}, and C = {2, 4, 6, 8, 10, 12}. Find the following: 1 A ∪ (B ∩ C) 2 A ∩ (B ∪ C) GE 112: MATHEMATICS IN THE MODERN WORLD Sets Sets (Application) Blood Group and Blood Types Karl Landsteiner won a Noble Prize in 1930 for his discovery of the four different human blood groups. He discovered that the blood of each individual contains exactly one of the following combination of antigens. a. Only A antigen (blood group A). b. Only B antigen (blood group B). c. Both A and B antigens (blood group AB). d. No A antigens and no B antigens (blood group O). GE 112: MATHEMATICS IN THE MODERN WORLD Sets Blood Group and Blood Types In 1941, Landsteiner and Alexander Wiener discov- ered that human blood may or may not contain Rh, or rhesus, factor. Blood with this factor is called Rh- positive and is denoted by Rh+, Blood without this factor is Rh-negative and is denoted by Rh-. GE 112: MATHEMATICS IN THE MODERN WORLD Sets Blood Group and Blood Types Use the venn diagrams to determine the blood type of each of the following people. a. Sue b. Lisa GE 112: MATHEMATICS IN THE MODERN WORLD Sets Example 3.9 A movie company is making plans for future movies it wishes to produce. The company has done a random survey of 1000 people. The results of the survey are shown below. 1 695 people like action adventures. 2 340 people like comedies. 3 180 people like both action adventures and comedies. Of the people surveyed, how many people a. like action adventures but not comedies? b. like comedies but not action adventures? c. do not like either of these types of movies? GE 112: MATHEMATICS IN THE MODERN WORLD Sets Principle of Inclusion and Exclusion (PIE) For all finite sets A and B, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). GE 112: MATHEMATICS IN THE MODERN WORLD Sets Principle of Inclusion and Exclusion (PIE) For all finite sets A and B, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Example 3.10 A school finds that 430 of its students are registered in chemistry, 560 are registered in mathematics, and 225 are registered in both chemistry and mathemat- ics. How many students are registered in chemistry or mathematics? GE 112: MATHEMATICS IN THE MODERN WORLD Sets Generalization of PIE For any n finite sets A1 , A2 ,... , An where n ≥ 2 n X X X |A1 ∪ A2 ∪ · · · ∪ An | = |Ai | − |Ai ∩ Aj | + |Ai ∩ Aj ∩ Ak | i=1 i