2-1: Logic with Inductive Reasoning Do NOW Identify the hypothesis and conclusion of each conditional. 1) A mapping that is a reflection is a type of transformation. 2) The quotient of two negative numbers is positive. 3) Determine if the conditional “If x is a number, then |x| > 0” is true. If false, give a counterexample. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Page 77 8) Roosevelt was inaugurated at age 42 10) Varied (x = –2) 11) 5 P.M. 12) 42 13) Hexagon 16) Approximately 26 students 20) Each term is the square of the previous term; 256, 65536 21) Each term is multiplied by ½; 1/16, 1/32 22) Each term is being multiplied by 3; alternating; –15, 18 23) 2n + 1 25) False 28) Approximately $400 31) 34, 55, 89; each term is the sum of the 2 previous terms 37) C 38) J 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Page 84 10) False, x = 2 and y = –4 12) Converse: If Brielle travels 10 miles in 20 minutes, then she drives exactly 30 mi/hr. - False Inverse: If Brielle does not drive at 30 mi/hr, then she does not travel 10 miles in 20 minutes – False Contrapositive: If Brielle does not travel 10 miles in 20 minutes, then she does not drive at 30 mi/hr - True 13) Hypothesis: An animal is a tabby Conclusion: It is a cat 14) Hypothesis: Two lines intersect Conclusion: Four angles are formed 15) Hypothesis: 8 oz. of cereal cost $2.99 Conclusion: 16 oz. of cereal cost $5.98 16) If a patient is ill, then you should monitor the patient’s heart rate 17) If the batter makes 3 strikes, then the batter is out 18) If segments are congruent, then they have equal measures 22) Converse: If an event is unlikely to occur, then the probability is 0.1 - FALSE Inverse: If the probability is not 0.1, then the event is likely to occur – FALSE Contrapositive: If an event is likely to occur, then the probability of the event is not 0.1 – TRUE 38) x = 5 42) If a mineral is calcite, then it has a hardness of 3 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

Deductive Reasoning – Page 7 Difference between Inductive vs Deductive Reasoning: Inductive Reasoning takes specific examples and makes general conclusions whereas Deductive Reasoning takes general information and breaks it down to specific information Process of using logic to draw conclusions from given facts, definitions, and properties. We use in proofs. Lawyers use in court cases. Law of Syllogism: If p →q and q → r are true, then p →r is true Denying the Consequent: If p →q and not q, then not r is true Law of Detachment: If p →q is true and p is true, then q is true 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

Inductive vs. Deductive Reasoning We use inductive reasoning to investigate and discover things about our world. Since the conjectures we make using our inductive reasoning is based on our fallible observation skills, we can be wrong. We can search for a counterexample to disprove our conjectures. In mathematics, we use our deductive reasoning to prove our conjectures beyond all uncertainty. In Inductive Reasoning, it does not always lead to the truth whereas Deductive Reasoning will always lead to the truth. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

Inductive vs Deductive Reasoning Inductive Reasoning Deductive Reasoning “Going Down” Specific Information General Information Specific Information General Information 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 1 In this statement, “An apple a day keeps the doctor away. Joe at an apple everyday. Dr. Dre stayed away” is a form of inductive or deductive reasoning? Deductive Reasoning 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 2 In this statement, “My dad has blond hair. My brother has blond hair. Therefore, everyone I am related to has blond hair” is a form of inductive or deductive reasoning? Inductive Reasoning 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 3 In this statement, “There is a myth that you can balance an egg on its end only on the spring equinox. A person was able to balance an egg on July 8, September 21, and December 19. Therefore this myth is false” is a form of inductive or deductive reasoning? Inductive Reasoning 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn In this statement, “Texas is a great state. The University of Texas is in Texas. So, University of Texas is a great state college.” is a form of inductive or deductive reasoning? Deductive Reasoning 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

Deductive Reasoning Explained Law of Detachment is Latin for “Modus Ponens” = Affirming the Antecedent Denying the Consequent is Latin for: “Modus Tollens” Law of Syllogism is the Chain Rule 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 4 Use the statements below to describe “Law of Detachment.” A: “If Mr. Dang had chalk on his fingers and then he had been playing billiards.” B: “Therefore, he has been playing billiards.” C: “If he had chalk between his fingers upon returning from the billiard place.” “If Mr. Dang had chalk on his fingers and then he had been playing billiards.” “If he had chalk between his fingers upon returning from the billiard place.” “Therefore, he has been playing billiards.” 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 5 Determine if the conjecture is valid by the Law of Detachment. “If a student passes his classes, the student is eligible to play sports. Ramon passed his classes.” Conjecture: Ramon is eligible to play sports. “If a student passes his classes, the student is eligible to play sports. Ramon passed his classes.” “If a student passes his classes.” “a student passes his classes.” The conjecture is valid. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 6 Determine if the conjecture is valid by the Law of Detachment. “I know if it is raining outside my workplace, then water is being added to my pool.” Conjecture: It is raining at my house. “I know if it is raining outside my workplace, then water is being added to my pool.” “If it is raining outside.” “Therefore, water is being added to the pool.” The conjecture is invalid. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn Determine if the conjecture is valid by the Law of Detachment. “In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series.” Conjecture: The Red Sox won the 2004 World Series. In the World Series, if a team wins four games, then the team wins the series. The conjecture is valid. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 7 Use the statements below to describe “Denying the Consequent”. A: “(Your religion) has not given us wings.” B: “If humans wanted to fly, (insert religion) would given us wings.” C: “(Your religion) did not want us to fly.” “If humans wanted to fly, (insert religion) would given us wings.” “(Your religion) has not given us wings.” “(Your religion) did not want us to fly.” 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 8 Use the statements below to describe “Law of Syllogism”. A: “If I have nightmares, then I will get very little sleep.” B: “If I eat pizza after midnight, then I will get very little sleep.” C: “If I eat pizza after midnight, then I will have nightmares.” “If I eat pizza after midnight, then I will have nightmares.” “If I have nightmares, then I will get very little sleep.” “If I have nightmares, then I will get very little sleep.” 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Example 9 Determine if the conjecture is valid by the Law of Syllogism. “If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon.” Conjecture: If a figure is a kite, then it is a polygon.” “If a figure is a kite, then it is a quadrilateral.” “If it is a quadrilateral, then the figure is a polygon.” “If a figure is a kite, then the figure is a polygon.” The conjecture is valid. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Your Turn Determine if the conjecture is valid by the Law of Syllogism. “If a number is divisible by 2, then it is even. If a number is even, then it is an integer. Conjecture: If a number is an integer, then it is divisible by 2.” “If a number is divisible by 2 then number is an integer.” “If a number is even, then number is an integer.” “If a number is an integer, then it is divisible by 2.” The conjecture is invalid. 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning

2-1: Logic with Inductive Reasoning Assignment Page 91 1, 4-12, 15-18, 23 5/27/2019 1:04 PM 2-1: Logic with Inductive Reasoning