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1. CHAPTER 2: MATHEMATICAL LANGUAGE AND SYMBOLS precise, concise, powerful GERLYN P. ORDONIO MMW INSTRUCTOR

2. LANGUAGE

3. Language is a system of communication that allows individuals to convey meaning using spoken or written symbols. It is a fundamental aspect of human society and plays a crucial role in various aspects of our lives. There are several forms of language, including spoken language, signed language, and symbolic language.

4. Is a system used in the field of mathematics to communicate mathematical ideas, concepts and theories among others. It is distinct and unique from the usual language that people are used to. Used to communicate abstract and logical ideas.

5. THREE CHARACTERISTICS OF MATHEMATICAL LANGUAGE PRECISE CONCISE POWERFUL

6. 3 Characteristics of mathematical language Mathematical language is able to make very fine distinction of things. 2 ≠ 22 b. means " accurate or exact " 1. Precise

7. 530 5:30 pm Five hundred thirty

8. Use to refer certain qualities concept ideas and others. b. The math symbols not only refer to different quantities but also represent the relationship between two quantities. All mathematical symbols are mainly used to perform mathematical operations under various concepts.

9. Mathematical language is able to say or write things briefly. We can convert mathematical language into expressions or equations. Ex. The sum of 2 and a number 2 + × Five time a number 5x 2. Concise

10. 3 Characteristics of mathematical language

11. Expression versus Sentences An expression (or mathematical expression ) is a finite combination of symbols that is well-defined according to rules that depend on the context. Symbols can designate numbers, variables, operations, functions, brackets, punctuations, and groupings to help determine order of operations, and other aspects of mathematical syntax. Expression – correct arrangement of mathematical symbols to represent the object of interest, does not contain a complete thought, and cannot be determined if it is true or false. Some types of expressions are numbers, sets, and functions.

12. Expression versus Sentences Sentence (or mathematical sentence ) – a statement about two expressions, either using numbers, variables, or a combination of both. Uses symbols or words like equals, greater than, or less than. It is a correct arrangement of mathematical symbols that states a complete thought and can be determined whether it’s true, false, sometimes true/sometimes false.

13. 1 + 6 = 7 1 + 6 = 8 x + 2 = 5 Sometimes true / Sometimes false sentence Tr ue if x = 3 and false otherwise.

14. Conventions in the Mathematical Language Mathematical Convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians. PEMDAS (Parenthesis, Exponent, Multiplication, Division, Addition and Subtraction.) All mathematical names and symbols are conventional.

15. Conventions in the Mathematical Language Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Formulas are written predominantly left to right, even when the writing system of the substrate language is right-to-left. Mathematical expressions = (equal) < (less-than) > (greater-than) + (addition) – (subtraction)  (multiplication)  (division)  (element)  (for all)  (there exists)  (infinity)  (implies)  (if and only if)  (approximately)  (therefore) Latin alphabet is commonly used for simple variables and parameters.

16. Four Basic Concepts Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Language of Sets Language of Functions Language of Relations Language of Binary Operations

17. Language of Sets Set theory is the branch of mathematics that studies sets or the mathematical science of the infinite. George Cantor (1845-1918) is a German Mathematician He is considered as the founder of set theory as a mathematical discipline.

18. Sets and Elements A set is a well-defined collection of objects. The objects are called the elements or members of the set.  element of a set  not an element of a set.

19. Some Examples of Sets D = { x x is an integer, 1  x  8} A = { x  x is a positive integer less than 10} B = { x x is a real number and x 2 – 1 = 0} C = { x x is a letter in the word dirt} E = { x x is a set of vowel letters} Set E equals the set of all x such that x is a set of vowel letters” or E = {a, e, i , o, u}

20. Indicate whether the ff. defined a Set a. The list of course offerings of Centro Escolar University. b. The elected district councilors of Manila City. c. The collection of intelligent monkeys in Manila Zoo. Answer: Answer: Answer: Set Set Not a set

21. List the Elements of the Sets a. A = { x  x is a letter in the word mathematics.} b. B = { x  x is a positive integer, 3  x  8.} c. C = { x  x = 2n + 3, n is a positive integer.} Answer: Answer: Answer: A = {m, a, t, h, e, i , c, s.} B = {3, 4, 5, 6, 7, 8} C = {5, 7, 9, 11, 13, …}

22. Methods of Writing Sets Roster Method . The elements of the set are enumerated and separated by a comma it is also called tabulation method. Rule Method . A descriptive phrase is used to describe the elements or members of the set it is also called set builder notation , symbol it is written as {x  P(x)}. Example: E = {a, e, i , o, u} E = { x  x is a collection of vowel letters} Roster method Rule method

23. Write the ff. Sets in Roster Form a. A= { x  x is the letter of the word discrete} b. B = {x 3  x  8, x  Z } c. C = { x x is the set of zodiac signs} Answer: A = { d, i , s, c, r, e, t} B = {4, 5, 6, 7} C = {Aries, Cancer, Capricorn, Sagittarius, Libra, Leo, …} Answer: Answer:

24. Write the ff. Sets using Rule Method a. D = { Narra , Mohagany , Molave , …} c. F = {Botany, Biology, Chemistry, Physics, …} b. E = {DOJ, DOH, DOST, DSWD, DENR, CHED, DepEd ,…} Answer: D = { x  x is the set of non-bearing trees.} E = { x  x is the set of government agencies. } F = { x  x is the set of science subjects. } Answer: Answer:

25. Some Terms on Sets  Finite and Infinite Sets.  Unit Set  Cardinality  Empty Set  Universal Set

26. Finite Set Finite set is a set whose elements are limited or countable, and the last element can be identified. Example: c. E = {a, e, i , o, u} b. C = {d, i , r, t} a. A = { x  x is a positive integer less than 10}

27. Infinite Set Infinite set is a set whose elements are unlimited or uncountable, and the last element cannot be specified. Example: c. H = { x  x is a set of molecules on earth} b. G = { x  x is a set of whole numbers} a. F = {…, –2, –1, 0, 1, 2,…}

28. Unit Set A unit set is a set with only one element it is also called singleton . Example: c. K = {rat} b. J = {w} a. I = { x  x is a whole number greater than 1 but less than 3}

29. Empty Set An empty set is a unique set with no elements (or null set ), it is denoted by the symbol  or { }. Example: c. N = { x  x is the set of positive integers less than zero} b. M = { x  x is a number of panda bear in Manila Zoo} a. L = { x  x is an integer less than 2 but greater than 1}

30. Universal Set Universal set is the all sets under investigation in any application of set theory are assumed to be contained in some large fixed set, denoted by the symbol U. Example: c. U = { x  x is an animal in Manila Zoo} b. U = {1, 2, 3,…,100} a. U = { x  x is a positive integer, x 2 = 4}

31. Cardinality The cardinal number of a set is the number of elements or members in the set, the cardinality of set A is denoted by n(A) Example: Determine its cardinality of the ff. sets c. C = {d, i , r, t} b. A = { x  x is a positive integer less than 10} a. E = {a, e, i , o, u}, n(E) = 5 n(A) = 9 n(C) = 4 Answer A = {1, 2, 3, 4, 5, 6, 7, 8, 9} Theorem 1.1: Uniqueness of the Empty Set : There is only one set with no elements.

32. Venn Diagram Venn Diagram is a pictorial presentation of relation and operations on set. Also known set diagrams , it show all hypothetically possible logical relations between finite collections of sets. Introduced by John Venn in his paper "On the Diagrammatic and Mechanical Representation of Propositions and Reasoning’s" Constructed with a collection of simple closed curves drawn in the plane or normally comprise of overlapping circles. The interior of the circle symbolically represents the elements (or members) of the set, while the exterior represents elements which are not members of the set.

33. Kinds of Sets  Subset  Proper Subset  Equal Set  Power Set Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

34. Subset If A and B are sets, A is called subset of B, if and only if, every element of A is also an element of B. Symbolically: A  B   x, x  A  x  B. Example: Suppose A = {c, d, e} B = {a, b, c, d, e} U = {a, b, c, d, e, f, g} Then A  B, since all elements of A is in B. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

35. Proper Subset Let A and B be sets. A is a proper subset of B, if and only if, every element of A is in B but there is at least one element of B that is not in A. Symbolically: A  B   x, x  A  x  B. Example: Suppose A = {c, d, e} B = {a, b, c, d, e} U = {a, b, c, d, e, f, g} Then A  B, since all elements of A is in B. C = {e, a, c, b, d} The symbol  denotes that it is not a proper subset. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

36. Equal Sets Given set A and B, A equals B, written, if and only if, every element of A is in B and every element of B is in A. Symbolically: A = B  A  B  B  A. Example: Suppose A = {a, b, c, d, e}, B = {a, b, d, e, c} U = {a, b, c, d, e, f, g} Then then A  B and B  A, thus A = B. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

37. Power Set Given a set S from universe U, the power set of S denoted by  (S), is the collection (or sets) of all subsets of S. Example: Determine the power set of (a) A = {e, f}, (b) = B = {1, 2, 3}. (a) A = {e, f}  (A) = {{e}, {f}, {e, f},  } (b) B = {1, 2, 3}  (B) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3},  }. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

38. Theorem Theorem 1.2: A Set with No Elements is a Subset of Every Set : If  is a set with no elements and A is any set, then   A. Theorem 1.3: For all sets A and B, if A  B then  (A)   (B). Theorem 1.4: Power Sets: For all integers n, if a set S has n elements then  (S) has 2 n elements. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

39. Operations on Sets  Union  Intersection  Complement  Difference  Symmetric Difference  Disjoint Sets  Ordered Pairs Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

40. Union The union of A and B, denoted A  B, is the set of all elements x in U such that x is in A or x is in B. Symbolically: A  B = { x  x  A  x  B}. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

41. Intersection The intersection of A and B, denoted A  B, is the set of all elements x in U such that x is in A and x is in B. Symbolically: A  B = { x  x  A  x  B}. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

42. Complement The complement of A (or absolute complement of A), denoted A’, is the set of all elements x in U such that x is not in A. Symbolically: A’ = {x  U  x  A}. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

43. Difference The difference of A and B (or relative complement of B with respect to A), denoted A  B, is the set of all elements x in U such that x is in A and x is not in B. Symbolically: A  B = { x  x  A  x  B} = A  B’. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

44. Symmetric Difference If set A and B are two sets, their symmetric difference as the set consisting of all elements that belong to A or to B, but not to both A and B. Symbolically: A  B = { x  x  (A  B)  x  (A  B)} = (A  B)  (A  B)’ or (A  B)  (A  B). Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

45. Example Suppose B = {c, d, e} U = {a, b, c, d, e, f, g} A = {a, b, c} Find the following a. A  B b. A B c. A’ d. A  B e. A  B Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

46. Solution a. A  B b. A  B d. A  B e. A  B = { a , b , c , d , e } = { c } = { a , b } = { a , b , d , e } c. A’ = {d, e, f, g } Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

47. Disjoint Sets Two set are called disjoint (or non-intersecting) if and only if, they have no elements in common. Symbolically: A and B are disjoint  A  B =  . Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

48. Order Pairs In the ordered pair (a, b), a is called the first component and b is called the second component . In general, (a, b)  (b, a). Example: Determine whether each statement is true or false. a. (2, 5) = (9 – 7, 2 + 3) b. {2, 5}  {5, 2} c. (2, 5)  (5, 2) Since 2 = 9 – 7 and 2 + 3 = 5, the ordered pair is equal. True Since these are sets and not ordered pairs, the order in which the elements are listed is not important. False These ordered pairs are not equal since they do not satisfy the requirements for equality of ordered pairs. True

49. Cartesian Product The Cartesian product of sets A and B, written AxB , is AxB = {(a, b)  a  A and b  B} Example: a. AxB Let A = {2, 3, 5} and B = {7, 8}. Find each set. b. BxA c. AxA = {(2, 7), (2, 8), (3, 7), (3, 8), (5, 7), (5, 8)} = {(7, 2), (7, 3), (7, 5), (8, 2), (8, 3), (8, 5)} = {(2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 3), (5, 5)}

50. Language of Functions and Relations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. A relation is a set of ordered pairs. If x and y are elements of these sets and if a relation exists between x and y , then we say that x corresponds to y or that y depends on x and is represented as the ordered pair of ( x , y ). A relation from set A to set B is defined to be any subset of A  B . If R is a relation from A to B and ( a , b )  R , then we say that “ a is related to b ” and it is denoted as a R b .

51. Language of Functions and Relations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Let A = { a , b , c , d } be the set of car brands, and B = { s , t , u , v } be the set of countries of the car manufacturer. Then A  B gives all possible pairings of the elements of A and B , let the relation R from A to B be given by R = {( a , s ), ( a , t ), ( a , u ), ( a , v ), ( b , s ), ( b , t ), ( b , u ), ( b , v ), ( c , s ), ( c , t ), ( c , u ), ( c , v ), ( d , s ), ( d , t ), ( d , u ), ( d , v )}.

52. Language of Functions and Relations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Let R be a relation from set A to the set B . domain of R is the set dom R dom R = { a  A  ( a , b )  R for some b  B }. image (or range) of R im R = { b  B  ( a , b )  R for some a  A }. Example: A = {4, 7}, Then A  A = {(4, 4), (4, 7), (7, 4),(7, 7)}. Let  on A be the description of x  y  x + y is even. Then (4, 4)   , and (7, 7)   .

53. Language of Functions and Relations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Function is a special kind of relation helps visualize relationships in terms of graphs and make it easier to interpret different behavior of variables.. Applications of Functions: financial applications economics medicine Engineering sciences natural disasters calculating pH levels measuring decibels designing machineries

54. Language of Binary Operations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. The set X is called the domain of the function. For each element of x in X , the corresponding element y in Y is called the value of the function at x , or the image of x . Range – set of all images of the elements of the domain is called the of the function. A function can map from one set to another.

55. Language of Binary Operations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Determine whether each of the following relations is a function. A = {(1, 3), (2, 4), (3, 5), (4, 6)} B = {(–2, 7), (–1, 3), (0, 1), (1, 5), (2, 5)} C = {(3, 0), (3, 2), (7, 4), (9, 1)}

56. Language of Binary Operations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Algebraic structures focuses on investigating sets associated by single operations that satisfy certain reasonable axioms. An operation on a set generalized structures as the integers together with the single operation of addition, or invertible 2  2 matrices together with the single operation of matrix multiplication. The algebraic structures known as group .

57. Binary Operations Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Let G be a set. A binary operation on G is a function that assigns each ordered pair of element of G . Symbolically, a  b = G , for all a , b , c  G .

58. Group Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. A group is a set of elements, with one operation, that satisfies the following properties: the set is closed with respect to the operation, the operation satisfies the associative property, there is an identity element, and each element has an inverse.

59. Group Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. A group is an ordered pair ( G ,  ) where G is a set and  is a binary operation on G satisfying the four properties. Closure property . If any two elements are combined using the operation, the result must be an element of the set. a  b = c  G , for all a , b , c  G . Associative property . ( a  b )  c = a  ( b  c ), for all a , b , c  G . Identity property . There exists an element e in G , such that for all a  G , a  e = e  a . Inverse property . For each a  G there is an element a –1 of G , such that a  a –1 = a –1  a = e .

60. Group Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. The set of group G contain all the elements including the binary operation result and satisfying all the four properties closure, associative, identity e , and inverse a –1 .

61. Example Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Determine whether the set of all non-negative integers under addition is a group. Solution: Apply the four properties to test the set of all non-negative integers under addition is a group. Step 1 : Closure property, choose any two positive integers, 8 + 4 = 12 and 5 + 10 = 15 The sum of two numbers of the set, the result is always a number of the set. Thus, it is closed.

62. Solution Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Step 2 : Associative property, choose three positive integers 3 + (2 + 4) = 3 + 6 = 9 (3 + 2) + 4 = 5 + 4 = 9 Thus, it also satisfies the associative property. Step 3 : Identity property, choose any positive integer 8 + 0 = 8; 9 + 0 = 9; 15 + 0 = 15 Thus, it also satisfies the identity property.

63. Solution Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Step 4 : Inverse property, choose any positive integer 4 + (–4) = 0; 10 + (–10) = 0; 23 + (–23) = 0 Note that a –1 = – a . Thus, it also satisfies the inverse property. Thus, the set of all non-negative integers under addition is a group, since it satisfies the four properties.

64. Formal Logic Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.  The science or study of how to evaluate arguments & reasoning.  It differentiate correct reasoning from poor reasoning.  It is important in sense that it helps us to reason correctly.  The methods of reasoning.

65. Mathematical Logic  Mathematical logic (or symbolic logic ) is a branch of mathematics with close connections to computer science. Four Divisions: Model Theory Set Theory Recursion Theory Proof Theory  It also study the deductive formal proofs systems and expressive formal systems.  Mathematical study of logic and the applications of formal logic to other areas of mathematics. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

66. Aristotle (382-322 BC) Aristotle is generally regarded as the Father of Logic The study started in the late 19th century with the development of axiomatic frameworks for analysis, geometry and arithmetic. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

67. Statement A statement (or proposition ) is a declarative sentence which is either true or false, but not both. The truth value of the statements is the truth and falsity of the statement. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

68. Example Which of the following are statements? 1. Manila is the capital of the Philippines. Is true A statement. 2. What day is it? It is a question Not a statement. 3. Help me, please. It cannot be categorized as true or false. Not a statement. 4. He is handsome. Is neither true nor false - “he” is not specified. Not a statement. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

69. Ambiguous Statements 2. Calculus is more interesting than Trigonometry. 3. It was hot in Manila. 4. Street vendors are poor. 1. Mathematics is fun. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

70. Propositional Variable A variable which used to represent a statement. A formal propositional written using propositional logic notation, p, q, and r are used to represent statements. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

71. Logical Connectives  A compound statement is a statement composed of two or more simple statements connected by logical connectives Logical connectives are used to combine simple statements which are referred as compound statements.  A statement which is not compound is said to be simple (also called atomic ). “exclusive-or.” “or” “if then” “and” “if and only if” “not” Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

72. Conjunction The conjunction of the statement p and q is the compound statement “p and q.” Symbolically, p  q , where  is the symbol for “ and .” Property 1: If p is true and q is true, then p  q is true; otherwise p  q is false. Meaning, the conjunction of two statements is true only if each statement is true. p q p  q T T F F T F T F T F F F Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

73. Example 1. 2 + 6 = 9 and man is a mammal. Determine the truth value of each of the following conjunction. 2. Manny Pacquiao is a boxing champion and Gloria Macapagal Arroyo is the first female Philippine President. 3. Ferdinand Marcos is the only three-term Philippine President and Joseph Estrada is the only Philippine President who resigns. False True False False True Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

74. Statement The disjunction of the statement p, q is the compound statement “p or q.” Property 2: If p is true or q is true or if both p and q are true, then p  q is true; otherwise p  q is false. Meaning, the disjunction of two statements is false only if each statement is false. Symbolically, p  q , where  is the symbol for “ or . ” p q p  q T T F F T F T F T T T F Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

75. Example Determine the truth value of each of the following disjunction. 1. 2 + 6 = 9 or Manny Pacquiao is a boxing champion. 2. Joseph Ejercito is the only Philippine President who resigns or Gloria Macapagal Arroyo is the first female Philippine President. 3. Ferdinand Marcos is the only three-term Philippine President or man is a mammal. False True True True True Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

76. Negation The negation of the statement p is denoted by  p , where  is the symbol for “ not .” Property 3: If p is true,  p is false. Meaning, the truth value of the negation of a statement is always the reverse of the truth value of the original statement. p  p T F F T Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

77. Example The following are statements for p, find the corresponding  p. 1. 3 + 5 = 8. 2. Sofia is a girl. 3. Achaiah is not here. 3 + 5  8. Sofia is a boy. Achaiah is here. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

78. Conditional The conditional (or implication ) of the statement p and q is the compound statement “if p then q.” Symbolically, p  q , where  is the symbol for “ if then .” p is called hypothesis (or antecedent or premise) and q is called conclusion (or consequent or consequence). Property 4: The conditional statement p  q is false only when p is true and q is false; otherwise p  q is true. Meaning p  q states that a true statement cannot imply a false statement. p q p  q T T F F T F T F T F T T Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

79. Example In the statement “If vinegar is sweet, then sugar is sour.” The antecedent is “ vinegar is sweet ,” and the consequent is “ sugar is sour .” Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

80. Example Obtain the truth value of each of the following conditional statements. 1. If vinegar is sweet, then sugar is sour. 2. 2 + 5 = 7 is a sufficient condition for 5 + 6 = 1. 3. 14 – 8 = 4 is a necessary condition that 6  3 = 2. True False True False False True False False True Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

81. Biconditional The biconditional of the statement p and q is the compound statement “p if and only if q.” Property 5: If p and q are true or both false, then p  q is true; if p and q have opposite truth values, then p  q is false. Symbolically, p  q , where  is the symbol for “ if and only if .” p q p  q T T F F T F T F T F F T Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

82. Example Determine the truth values of each of the following biconditional statements. 1. 2 + 8 = 10 if and only if 6 – 3 = 3. 2. Manila is the capital of the Philippines is equivalent to fish live in moon. 3. 8 – 2 = 5 is a necessary and sufficient for 4 + 2 = 7. True False True True True Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

83. Exclusive-Or The exclusive-or of the statement p and q is the compound statement “p exclusive or q.” Property 6: If p and q are true or both false, then p  q is false; if p and q have opposite truth values, then p  q is true. Symbolically, p  q , where  is the symbol for “ exclusive or .” p q p  q T T F F T F T F F T T F Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

84. Example “Sofia will take her lunch in Batangas or she will have it in Singapore.” Case 1 : Sofia cannot have her lunch in Batangas and at the same time in do it in Singapore,” Case 2 : If Sofia will have her lunch in Batangas or in Singapore, meaning she can only have it in one location given a single schedule. Case 3 : If she ought to decide to have her lunch elsewhere (neither in Batangas nor in Singapore). False True False Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

85. Predicate A predicate (or open statements ) is a statement whose truth depends on the value of one or more variables. Predicates become propositions once every variable is bound by assigning a universe of discourse . Most of the propositions are define in terms of predicates “ x is an even number” is a predicate whose truth depends on the value of x . Example: Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

86. Predicate A predicate can also be denoted by a function-like notation. P ( x ) = “ x is an even number.” Now P (2) is true, and P (3) is false. If P is a predicate, then P ( x ) is either true or false, depending on the value of x . Example: Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

87. Propositional Function A propositional function is a sentence P(x); it becomes a statement only when variable x is given particular value. Propositional functions are denoted as P(x), Q(x),R(x), and so on. The independent variable of propositional function must have a universe of discourse, which is a set from which the variable can take values. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

88. Propositional Function “If x is an odd number, then x is not a multiple of 2.” The given sentence has the logical form P(x)  Q(x) and its truth value can be determine for a specific value of x. There exists an x such that x is odd number and 2x is even number. Example: For all x, if x is a positive integer, then 2x + 1 is an odd number. Example: Existential Quantifiers Universal Quantifiers Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

89. Universe of Discourse The universe of discourse for the variable x is the set of positive real numbers for the proposition “There exists an x such that x is odd number and 2x is even number.” Binding variable is used on the variable x, we can say that the occurrence of this variable is bound . A variable is said to be free , if an occurrence of a variable is not bound. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

90. Universe of Discourse To convert a propositional function into a proposition, all variables in a proposition must be bound or a particular value must be designated to them. The scope of a quantifier is the part of an assertion in which variables are bound by the quantifier. This is done by applying combination of quantifiers (universal, existential) and value assignments. A variable is free if it is outside the scope of all quantifiers. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

91. Existential Quantifiers The statement “there exists an x such that P(x),” is symbolized by  x P(x) . The symbol  is called the existential quantifier The statement “  x P(x) ” is true if there is at least one value of x for which P(x) is true. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

92. Universal Quantifiers The statement “ for all x, P(x) ,” is symbolized by  x P(x) . The symbol  is called the universal quantifier . The statement “  x P(x) ”is true if only if P(x) is true for every value of x. Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.

93. Topic Outline Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Quantifier Symbol Translation Existential  There exists There is some For some For which For at least one Such that Satisfying Universal  For all For each For every For any Given any

94. Truth Values of Quantifiers Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. If the universe of discourse for P is P{p 1 , p 2 , …, p n }, then  x P( x )  P(p 1 )  P(p 2 )  …  P( p n ) and  x P( x )  P(p 1 )  P(p 2 )  …  P( p n ). Statement Is True when Is False when  x P(x) P(x) is true for every x. There is at least one x for which P(x) is false.  x P(x) There is at least one x for which P(x) is true. P(x) is false for every x.

95. Quantified Statements and their Negation Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D. Statement Negation All A are B. Some A are not B. No A are B. Some A are B. Some A are not B. All A are B. Some A are B. No A are B.

96. For the things of this world cannot be made known without a knowledge of mathematics. – Roger Bacon Copyright 2018: Mathematics in the Modern World by Winston S. Sirug , Ph.D.