MATHEMATICS LOGIC AND SET THEORY PRESENTATION.pptx

Mathematics major for licensure examination for professional teachers.

1. Hello

2. LOGIC SET THEORY &

3. LOGIC

4. Gottfried Wilhelm Leibniz tries to advance the study of logic from a mere philosophical subject to a formal mathematical subject. Leibniz never completely achieved this goal; however, several mathematicians, such as Augustus de Morgan and George Boole , contributed to the advancement of symbolic logic as a mathematical discipline .

5. LOGIC Logic is the study of the methods and principles of reasoning.

6. LOGIC STATEMENT A logic statement or proposition is a declarative sentence that is true or false but not both. There must be no ambiguity. In logic, the truth of a statement is established beyond ANY doubt by a well-reasoned argument.

7. LOGIC STATEMENT Examples: You will pass the licensure examination for teachers. You will be a topnotcher. You did well today. It’s okay not to be okay. Loving him was red.

8. LOGIC STATEMENT Exercise: Determine whether each sentence is a statement. Do you think you'll pass the LEPT? I love Philippines. Wena is a good dancer. Did he cheat on Kath? Please give me another chance.  

9. SIMPLE STATEMENT A simple statement is a statement that conveys a single idea. Examples: a) Zero times any real number is zero. b) 1+1=2. c) All birds can fly

10. COMPOUND STATEMENT A compound statement is a statement that conveys two or more ideas. It contains several simple statements. The ideas in a compound statement are connected by connectives.

11. Mathematical statements may be joined by logical connectives, such as and, or, if . . . then, and if and only if, which are used to combine simple propositions to form compound statements. These connectives are negation, conjunction, disjunction, implication, and biconditional. LOGICAL CONNECTIVES

12. Examples: The grass is green and the sky is blue. It is cold or it is sunny. If a person is kind, then he is helpful. The number 12 is an even number if and only if it is divisible by 2. LOGICAL CONNECTIVES

13. Statements can be represented by propositional variables 𝒑, 𝒒. LOGICAL CONNECTIVES LOGIC SYMBOLS NOTATION MEANING Negation Conjunction Disjunction Conditional / Implication Biconditional LOGIC SYMBOLS NOTATION MEANING Negation Conjunction Disjunction Conditional / Implication Biconditional

14. The negation of a statement is the opposite of a given mathematical statement. Examples: 2 is the smallest prime number. I am feeling well tonight. I am cute. NEGATION OF STATEMENT 2 is not the smallest prime number. I am not feeling well tonight. I am not cute.

15. WRITING COMPOUND STATEMENTS IN SYMBOLIC FORM Consider the following simple statements. p: Today is Tuesday. q: It is raining. r: I am going to a movie date. s: I am not going to a basketball game. Write the following compound statements in symbolic form. Today is Tuesday and it is raining.  

16. WRITING COMPOUND STATEMENTS IN SYMBOLIC FORM Consider the following simple statements. p: Today is Tuesday. q: It is raining. r: I am going to a movie date. s: I am not going to a basketball game. Write the following compound statements in symbolic form. It is raining and I am going to a movie date.  

17. WRITING COMPOUND STATEMENTS IN SYMBOLIC FORM Consider the following simple statements. p: Today is Tuesday. q: It is raining. r: I am going to a movie date. s: I am not going to a basketball game. Write the following compound statements in symbolic form. I am going to the basketball game or I am going to a movie date.  

18. WRITING COMPOUND STATEMENTS IN SYMBOLIC FORM Consider the following simple statements. p: Today is Tuesday. q: It is raining. r: I am going to a movie date. s: I am not going to a basketball game. Write the following compound statements in symbolic form. If it is raining, then I am not going to the basketball game.  

19. TRANSLATE SYMBOLIC STATEMENTS Consider the following simple statements. p: The pageant will be held in Manila. q: The pageant will be televised on ABS-CBN. r: The pageant will not be shown in GMA. s: The Philippines' candidate is favored to win. Write the following compound statements in symbolic form. The pageant will be televised on ABS-CBN and it will be held in Manila.  

20. TRANSLATE SYMBOLIC STATEMENTS Consider the following simple statements. p: The pageant will be held in Manila. q: The pageant will be televised on ABS-CBN. r: The pageant will not be shown in GMA. s: The Philippines' candidate is favored to win. Write the following compound statements in symbolic form. The pageant will be shown in GMA and the Philippines' candidate is favored to win.  

21. TRANSLATE SYMBOLIC STATEMENTS Consider the following simple statements. p: The pageant will be held in Manila. q: The pageant will be televised on ABS-CBN. r: The pageant will not be shown in GMA. s: The Philippines' candidate is favored to win. Write the following compound statements in symbolic form. If t he Philippines' candidate is favored to win, then the pageant will be held in Manila. .  

22. A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. Every conditional statement has three related statements. For every implication or conditional statement (𝑝→𝑞), we can construct its converse, inverse, and contrapositive. CONVERSE, INVERSE, & CONTRAPOSITIVE

23. CONVERSE To form the converse of the conditional statement (𝑝→𝑞), interchange the hypothesis and the conclusion. (𝑞→ 𝑝)

24. INVERSE To form the inverse of the conditional statement (𝑝→𝑞), take the negation of both the hypothesis and the conclusion.  

25. CONTRAPOSITIVE To form the contrapositive of the conditional statement (𝑝→𝑞), interchange the hypothesis and the conclusion of the inverse statement.  

26. Example: If I get a job, then I can help my parents. Converse: If I can help my parents, then I get a job. Inverse: If I don’t get a job, then I cannot help my parents. Contrapositive: If I can’t help my parents, then I won’t get a job. CONVERSE, INVERSE, & CONTRAPOSITIVE

27. TRUTH TABLES, TAUTOLOGIES, LOGICAL EQUIVALENCES &

28. Mathematicians normally use a two-valued logic : Every statement is either True or False. This is called the Law of the Excluded Middle . A statement in sentential logic is built from simple statements using the logical connectives , , , , and . The truth or falsity of a statement built with these connectives depends on the truth or falsity of its components.  

29. A truth table shows how the truth or falsity of a compound statement depends on the truth or falsity of the simple statements from which it's constructed.

30. TRUTH TABLE FOR NEGATION If P is true, its negation is false. If P is false, then is true.   T F F T T F F T

31. TRUTH TABLE FOR CONJUNCTION should be true when both P and Q are true, and false otherwise.   T T T T F F F T F F F F T T T T F F F T F F F F

32. TRUTH TABLE FOR DISJUNCTION is true if either P is true or Q is true or both. It's only false if both P and Q are false.   T T T T F T F T T F F F T T T T F T F T T F F F

33. TRUTH TABLE FOR CONDITIONAL The statement “if P then Q” is true if both P and Q are true, or if P is false. T T T T F F F T T F F T T T T T F F F T T F F T

34. TRUTH TABLE FOR BICONDITIONAL means that P and Q are equivalent. So, the double implication is true if P and Q are both true or if P and Q are both false; otherwise, the double implication is false.   T T T T F F F T F F F T T T T T F F F T F F F T

35. THINGS TO REMEMBER When constructing a truth table, do consider all possible assignments of True (T) and False (F) to the component statements. Each of these statements can be either true or false, so there are possibilities.  

36. THINGS TO REMEMBER To avoid duplication or omission in assigning truth values to the component statements, the easiest and most systematic approach is to use lexicographic ordering .

37. THINGS TO REMEMBER Example: For a compound statement with three components P, Q, and R, here are the possible assignments:

38. THINGS TO REMEMBER There are different ways of setting up truth tables. For instance, write the truth values "under" the logical connectives of the compound statement, gradually building up to the column for the "primary" connective.

39. EXAMPLE Construct a truth table for the compound statement :   T T F T F T F F F F F T T T T F F T T T T T F T F T F F F F F T T T T F F T T T

40. TAUTOLOGY A tautology is a formula that is "always true“, that is, it is true for every assignment of truth values to its simple components. Think of tautology as a rule of logic.

41. CONTRADICTION A contradiction is false for every assignment of truth values to its simple components.

42. EXAMPLE Show that is a tautology.   T T T T T T F F T T F T T F T F F T T T T T T T T T F F T T F T T F T F F T T T

43. LOGICALLY EQUIVALENT Two statements X and Y are logically equivalent if is a tautology. Another way to say this is: For each assignment of truth values to the simple statements that make up X and Y, the statements X and Y have identical truth values.  

44. EXAMPLE Show that and are logically equivalent.   T T T F T T F F F F F T T T T F F T T T T T T F T T F F F F F T T T T F F T T T This tautology is called Conditional Disjunction. You can use this equivalence to replace a conditional by a disjunction.

45. TAUTOLOGIES & LOGICAL EQUIVALENCES When a tautology has the form of a biconditional, the two statements that make up the biconditional are logically equivalent. Hence, you can replace one side with the other without changing the logical meaning.

46. SET THEORY

47. SET A SET is a collection of well-defined objects. The objects in the set are called the ELEMENTS of the set. To describe a set, we use braces { }, and use capital letters to represent it. To indicate membership, we use the symbol ∈, when an element is not a membership, we use  .

48. SET Examples: A = {2, 4, 6, 8, 10} B = {all licensed professional teachers} C = { } D = {consonants of the English alphabet} E = {Instagram, Facebook, X, TikTok} F = {x ∈ N | x < 5}  

49. SET REPRESENTATION

50. SET REPRESENTATION Recursive Rule By defining a set of rules which generates or defines its members. Examples: B = {all licensed professional teachers} D = {consonants of the English alphabet}

51. SET REPRESENTATION Listing / Roster Method Writing or listing down all the elements between braces. Examples: A = {2, 4, 6, 8, 10} E = {Instagram, Facebook, X, TikTok}

52. SET REPRESENTATION Set-Builder Notation Enumerating its elements by stating the properties that its members must satisfy. Examples: C = { } F =  

53. TYPES SETS of

54. A finite set contains elements that can be counted and terminates at a certain natural number. Examples of Finite Set: A = {2, 4, 6, 8, 10} D = {consonants of the English alphabet} E = {Instagram, Facebook, X, TikTok} F = {x ∈ N | x < 5} FINITE SET

55. An infinite set is a set whose elements can not be counted. An infinite set is one that has no last element. Examples of Infinite Set: B = {all licensed professional teachers} C = { }   INFINITE SET

56. This is a set with no elements, often symbolized by ∅ or { }. Examples: G = {vowel in the word “CRYPT”} G = ∅ NULL SET

57. A set with only one member. Examples: H = {number that is an even prime number} H = {2} SINGLETON SET

58. Two sets are equal if they contain the same elements. Examples: F = {x ∈ N | x < 5} I= {1, 2, 3, 4} EQUAL SETS

59. Two sets are equivalent if they contain the same number of elements. Example: E = {Instagram, Facebook, X, TikTok} F = {x ∈ N | x < 5} EQUIVALENT SETS

60. A set that contains all the elements considered in a particular situation and denoted by U. Example: U = {letters of the English alphabet} J = { b , c , d , e , f , g , h , i , j } UNIVERSAL SET

61. A set A is called a subset of B if every element of A is also an element of B. “A is a subset of B” is written as A  B. ∅ is a subset of every set. A set is always a subset of itself. SUBSET

62. Example: I = { 1, 2, 3, 4} Subsets: { 1 }, { 2 }, { 3 }, { 4 }, { 1, 2 }, { 1, 3 }, { 1, 4 }, { 2, 3 }, { 2, 4 }, { 3, 4 }, { 1 , 2 , 3 }, { 1 , 2 , 4 }, { 1 , 3 , 4 }, { 2 , 3 , 4 }, { 1 , 2 , 3 , 4 }, and ∅ SUBSET

63. This is defined to be the set of all subsets of a given set, written as P(A). Example: I = { 1 , 2 , 3 , 4 } P (L) = { { 1 } , { 2 } , { 3 } , { 4 } , { 1 , 2 } , { 1 , 3 } , { 1 , 4 }, { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 1 , 2 , 3 } , { 1 , 2 , 4 }, { 1 , 3 , 4 } , { 2 , 3 , 4 }, { 1 , 2 , 3 , 4 }, ∅ }   elements / subsets in the .   POWER SET

64. Two sets are disjoint if they have no element in common. Example: D = { consonants of the English alphabet } K = { vowels of English alphabet } Sets K and C are disjoint since they do not have elements in common. DISJOINT SETS

65. CARDINALITY OF THE SET The cardinality of a set is its size. For a finite set, the cardinality of a set is the number of members it contains. In symbolic notation the size of a set S is written |S|.

66. AXIOM OF EXTENSION This states that a set is completely determined by what its elements are – not the order in which they might be listed or the fact that some elements might be listed more than once.

67. AXIOM OF EXTENSION Through the Axiom of Extension, sets can be written not like this: × L = { a , b , b , c , d , e , e , f , g , h , h , i , j } But can be written like any of these: L = { a , b , c , d , e , f , g , h , i , j } L= { j , g , c , a , e , b , h , f , i , d }

68. SET OPERATIONS

69. The English logician John Venn (1834–1923) developed diagrams, which we now refer to as Venn diagrams, that can be used to illustrate sets and relationships 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. In a Venn Diagram, the size of the rectangle or circle is not a concern. VENN DIAGRAMS

70. UNION The union of sets A and B, denoted by A∪B, is the set consisting of all elements that belong to either A or B or both.  

71. UNION Example for union of sets: M = {1, 2, 3, 4, 5, 6, 7, 11} J = {2, 4, 6, 8, 10, 12} M ∪ J = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12}  

72. INTERSECTION The intersection of sets A and B, denoted by A∩B, is the set consisting of all elements that belong both A and B.  

73. INTERSECTION Example for intersection of sets: M = {1, 2, 3, 4, 5, 6, 7, 11} J = {2, 4, 6, 8, 10, 12} M ∩ J = {2, 4, 6}  

74. COMPLEMENT The complement of set A is defined as the set consisting of all elements in U that are not in A.  

75. COMPLEMENT Example: U = { x ∈ N | x < 21} J = {2, 4, 6, 8, 10, 12} J' = { 1, 3, 5, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20}  

76. DIFFERENCE The difference or relative complement of two sets A and B, denoted by A–B, is the set consisting of all elements in A that are not in B.  

77. DIFFERENCE The difference or relative complement of two sets A and B, denoted by A–B, is the set consisting of all elements in A that are not in B.  

78. DIFFERENCE Example for difference of sets: M = {1, 2, 3, 4, 5, 6, 7, 11} J = {2, 4, 6, 8, 10, 12} M J = {1, 3, 5, 7, 11} J M = {8, 10, 12 }    

79. RATIONALIZE !

80. PASSED OR SHOW YOUR TALENT ?

81. 1 A   Write a negation for the statement:   B   C   D  

82. 2 A It is not the case that students are happy or teachers are not happy . Translate the symbolic compound statement into words. Let represent the statement: "Students are happy" and let represent the statement: "Teachers are happy."   B Students are not happy and teachers are not happy. C It is not the case that students are happy and teachers are not happy. D Students are not happy or teachers are not happy.

83. 3 A We may not deduct points if there is no work to justify your answer. Write the negation of the statement: We will deduct points if there is no work to justify your answer. If not a statement, state so. B We will not deduct points if there is no work to justify your answer. C We will not deduct points if there is work to justify your answer D The excerpt is not a statement.

84. 4 A Do not make sure that you fill in the circle on the answer sheet that corresponds to your answer choice. Write the negation of the statement: Make sure that you fill in the circle on the answer sheet that corresponds to your answer choice. If not a statement, state so. B Do not make sure that you do not fill in the circle on the answer sheet that corresponds to your answer choice. C Make sure that you do not fill in the circle on the answer sheet that corresponds to your answer choice. D The excerpt is not a statement.

85. 5 A   Give the number of rows in the truth table for the compound statement:   B   C   D  

86. SOLUTION We have 5 logical statements: p, q, r, s, and t.        

87. 5 A   Give the number of rows in the truth table for the compound statement:   B   C   D  

88. 6 A   Find the number of subsets of the set:   B   C   D  

89. SOLUTION We have 5 elements in the given set.        

90. 6 A   Find the number of subsets of the set:   B   C   D  

91. 7 A 6   Find the number of subsets of the set:   B   C   D  

92. SOLUTION We have 10 elements in the given set: {18, 20, 22, 24, 26, 28, 30, 32, 34, 36}      

93. 7 A 6   Find the number of subsets of the set:   B   C   D  

94. 8 Construct a truth table for the compound statement:  

95. SOLUTION Construct a truth table for the compound statement:   T T F F F T F F T F F T T F F F F T T T T T F F F T F F T F F T T F F F F T T T

96. 8 Construct a truth table for the compound statement:  

97. 9 Construct a truth table for the compound statement:  

98. SOLUTION Construct a truth table for the compound statement:   T T T F T T F T F T F T T F T F F F T F T T T F T T F T F T F T T F T F F F T F

99. 9 Construct a truth table for the compound statement:  

100. 10 Construct a truth table for the compound statement:  

101. SOLUTION Construct a truth table for the statement:   ) T T T F T T T F F F F T F T T T T T F F T T T T T T T F T T T F F F F T F T T T T T F F T T T T

102. 10 Construct a truth table for the compound statement:  

103. 11 Construct a truth table for the compound statement:  

104. SOLUTION Construct a truth table for the statement:   T T T F F F T T F F T T T T F T T F F F T F F T F T F T T T T F F F T T F F T T T T F T T F F F T F F T F T F T

105. 11 Construct a truth table for the compound statement:  

106. 12 A {t, v, x} Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. Find .   B   C   D  

107. SOLUTION Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. Find .   U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z}    

108. 12 A {t, v, x} Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. Find .   B   C   D  

109. 13 A {r, t, u} Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. Find .   B   C   D  

110. SOLUTION Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. Find .   U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z} C = {v, w, x, y, z}        

111. 13 A {r, t, u} Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. Find .   B   C   D  

112. 14 A     , and ; what is ?   B   C   D  

113. SOLUTION   , and ; what is ?     A B 2 2 7  

114. 14 A     , and ; what is ?   B   C   D  

115. 15 Shade the Venn diagram to represent the set: A' ∩ B' A B C D

116. SOLUTION A' ∩ B' A B A' A B B' A B

117. 15 Shade the Venn diagram to represent the set: A' ∩ B' A B C D

118. 16 Shade the Venn diagram to represent the set:   A B C D

119. SOLUTION (𝐴 ∩ 𝐵) ∪ (𝐴 ∪ 𝐵)′ A B 𝐴 ∩ 𝐵 A B 𝐴 ∪ 𝐵 A B (𝐴 ∪ 𝐵)’ A B

120. 16 Shade the Venn diagram to represent the set:   A B C D

121. 17 A 48 Use a Venn Diagram and the given information to determine the number of elements in the indicated region. If and , find .   B   C   D  

122. SOLUTION If 𝑛(𝑈) = 60, 𝑛(𝐴) = 34, 𝑛(𝐵) = 22, and 𝑛(𝐴 ∩ 𝐵) = 8, find 𝑛(𝐴 ∪ 𝐵)′. A B 8 26 14 12

123. 17 A 48 Use a Venn Diagram and the given information to determine the number of elements in the indicated region. If and , find .   B   C   D  

124. 18 A 7 If , , , , , , , and , Find .   B   C   D  

125. SOLUTION 𝑛(𝐴 ∪ 𝐵 ∪ 𝐶) = 77 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶) = 11 𝑛(𝐴 ∩ 𝐵) = 24 𝑛(𝐴 ∩ 𝐶) = 21 𝑛(𝐵 ∩ 𝐶) = 19 𝑛(𝐴) = 56 𝑛(𝐵) = 38 𝑛(𝐶) = 36 Find 𝑛(𝐴′ ∩ 𝐵 ∩ 𝐶). A B 26 C 11 13 10 8 22 6 7

126. 18 A 7 If , , , , , , , and , Find .   B   C   D  

127. 19 A 19 A survey of 240 families showed that 91 had a dog; 70 had a cat; 31 had a dog and a cat; 91 had neither a cat nor a dog, and in addition did not have a parakeet; 7 had a cat, a dog, and a parakeet. How many had a parakeet only? B   C   D  

128. SOLUTION 91 had a dog 70 had a cat 31 had a dog and a cat 91 had neither a cat nor a dog, and in addition did not have a parakeet 7 had a cat, a dog, and a parakeet. C D P 7 24 91 60 39

129. 19 A 19 A survey of 240 families showed that 91 had a dog; 70 had a cat; 31 had a dog and a cat; 91 had neither a cat nor a dog, and in addition did not have a parakeet; 7 had a cat, a dog, and a parakeet. How many had a parakeet only? B   C   D  

130. 20 A True Determine whether the statement is true or false.   B False

131. 21 A True Determine whether the statement is true or false.   B False

132. 22 A True Determine whether the statement is true or false.   B False

133. 23 A True Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false.   B False

134. 24 A True Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false.   B False

135. 25 A True Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false.   B False

136. 26 A True Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. C ⊈ B B False

137. 27 A True Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. A ⊂ A B False

138. 28 A True Let represent a true statement, and let and represent false statements. Find the truth value of the given compound statement:   B False

139. SOLUTION Find the truth value of the given compound statement: ~(~𝑝 ∧ ~𝑞) ∨ (~𝑟 ∨ ~𝑝) T F F F T T F T T T F F F T T F T T T T

140. 28 A True Let represent a true statement, and let and represent false statements. Find the truth value of the given compound statement:   B False

141. 29 A   Are the statements equivalent?   B   C   D  

142. SOLUTION Are the statements ~(𝑞 → 𝑝) 𝑎𝑛𝑑 𝑞 ∧ ~𝑝 equivalent? T T T F F F T F F T T F F T T F F F F F T F T F T T T F F F T F F T T F F T T F F F F F T F T F

143. 29 A   Are the statements equivalent?   B   C   D  

144. 30 A   Are the statements equivalent?   B   C   D  

145. SOLUTION Are the statements 𝑞 ∧ ~𝑝 𝑎𝑛𝑑 ~𝑝 → ~𝑞 equivalent? T T F F F T T F T T F F F T F F T T F F T F T T T T F F F T T F T T F F F T F F T T F F T F T T

146. 30 A   Are the statements equivalent?   B   C   D  

147. 31 A True Let p represent a true statement, and let q represents a false statement. Find the truth value of the compound statement:   B False

148. SOLUTION Find the truth value of the compound statement:   T F F F T F T T F F F T F T

149. 31 A True Let p represent a true statement, and let q represents a false statement. Find the truth value of the compound statement:   B False

150. 32 A The sun does not come out tomorrow and the roses will not open. Write the statement, “If the sun comes out tomorrow, the roses will open,” as an equivalent statement that does not use the if . . . then connective. (Remember that is equivalent to .)   B The sun does not come out tomorrow or the roses will not open. C The sun does not come out tomorrow or the roses will open. D The sun comes out tomorrow and the roses will not open.

151. 33 A If it is raining, you do not take your umbrella. Write the negation of the statement: If it is raining, you take your umbrella. (Remember that the negation of p → q is p ∧ ~q.) B It is raining and you do not take your umbrella. C It is not raining and you do not take your umbrella. D It is not raining and you take your umbrella.

152. 34 A 7x + 2y > -3, so the answer is not "Lake ". Write the negation of the statement: If 7x + 2y > 3, the answer is "Lake ". B 7x + 2y ≤ 3 and the answer is not "Lake ". C If 7x + 2y > 3, the answer is not "Lake ". D 7x + 2y > 3 and the answer is not "Lake ".

153. 35 A Collection of objects A set is: B Well-defined collection of objects C Elements of the set D Description

154. 36 A 106 In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics and chemistry, 22 had taken chemistry and physics course, and 14 had taken all the three courses. Find how many had taken one course only. B   C   D  

155. SOLUTION 64 had taken math 94 had taken chemistry 58 had taken physics 28 had taken math & physics 26 had taken math & chem 22 had taken chem & physics 14 had taken all C P M 14 8 12 14 22 60 24

156. 36 A 106 In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics and chemistry, 22 had taken chemistry and physics course, and 14 had taken all the three courses. Find how many had taken one course only. B   C   D  

157. 37 A 10 An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. How many uses only Radio? B   C   D  

158. SOLUTION 115 use Television 110 use Radio 130 use Magazines 85 use TV & Magazines 75 use TV & Radio 95 use Radio & Magazines 70 use all the three T R M 70 5 15 25 10 25 20

159. 37 A 10 An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. How many uses only Radio? B   C   D  

160. 38 A 10 An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. How many uses only Television? B   C   D  

161. SOLUTION 115 use Television 110 use Radio 130 use Magazines 85 use TV & Magazines 75 use TV & Radio 95 use Radio & Magazines 70 use all the three T R M 70 5 15 25 10 25 20

162. 38 A 10 An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. How many uses only Television? B   C   D  

163. 39 A 10 An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. How many uses both Television and Magazine but not radio? B   C   D  

164. SOLUTION 115 use Television 110 use Radio 130 use Magazines 85 use TV & Magazines 75 use TV & Radio 95 use Radio & Magazines 70 use all the three T R M 70 5 15 25 10 25 20

165. 39 A 10 An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. How many uses both Television and Magazine but not radio? B   C   D  

166. 40 A 8 There are 30 students in a class. Among them, 8 students are learning both English and French. A total of 18 students are learning English. If every student is learning at least one language, how many students are learning French in total? B   C   D  

167. SOLUTION 30 students in a class 8 students are learning both English and French 18 students are learning English every student is learning at least one language E F 8 10 12

168. 40 A 8 There are 30 students in a class. Among them, 8 students are learning both English and French. A total of 18 students are learning English. If every student is learning at least one language, how many students are learning French in total? B   C   D  

169. Thank You