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2.1 Use Inductive Reasoning

2.1 Use Inductive Reasoning. Objectives To form conjectures through inductive reasoning To disprove a conjecture with a counterexample To avoid fallacies of inductive reasoning. Example 1.

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2.1 Use Inductive Reasoning

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  1. 2.1 Use Inductive Reasoning Objectives • To form conjectures through inductive reasoning • To disprove a conjecture with a counterexample • To avoid fallacies of inductive reasoning

  2. Example 1 You’re at school eating lunch. You ingest some air while eating, which causes you to belch. Afterward, you notice a number of students staring at you with disgust. You burp again, and looks of distaste greet your natural bodily function. You have similar experiences over the course of the next couple of days. Finally, you conclude that belching in public is socially unacceptable. The process that lead you to this conclusion is called inductive reasoning.

  3. Inductive Reasoning Inductive reasoning is the process of observing data, recognizing patterns, and making generalizations based on your observations.

  4. Generalization Generalization: statement that applies to every member of a group • Science = hypothesis • Math = conjecture

  5. Conjecture A conjecture is a general, unproven statement believed to be true based on investigation or observation

  6. Inductive Reasoning Inductive reasoning can be used to make predictions about the future based on the past or to make conjectures about the past based on the present.

  7. Example 2 A scientist takes a piece of table salt, turns it over a Bunsen burner, and observes that it burns with a yellow flame. She does this with many other pieces of table salt, finding they all burn with a yellow flame. She therefore makes the conjecture: “All table salt burns with a yellow flame.”

  8. Inductive Reasoning Inductive Reasoning General Specific

  9. Example 3 Find the 12th number in the following sequence: 1, 1, 2, 3, 5, 8, … This is called the Fibonacci Sequence!

  10. Example 4 Numbers such as 3, 4, and 5 are consecutive numbers. Make and test a conjecture about the sum of any three consecutive numbers. Answer in your notebook

  11. Example 5 Use the map of Texas provided to formulate a conjecture about the numbering system of Interstate Highways. Answer in your notebook

  12. Example 6 (An allegory) Student A neglected to do his/her homework on numerous occasions. When Student A's mean teacher popped a quiz on the class, Student A failed. After the quiz, Student A had several other HW assignments that he/she also neglected to complete. When test time rolled around, Student A failed the exam . Students B-F behaved in a similar, academically deplorable manner. Use inductive reasoning to make a conjecture about the relationship between homework and test/quiz performance. Answer in your notebook

  13. Example 7 Inductive reasoning does not always lead to the truth. What are some famous examples of conjectures that were later discovered to be false?

  14. To Prove or To Disprove In science, experiments are used to prove or disprove an hypothesis. In math, deductive reasoning is used to prove conjectures and counterexamples are used to disprove them.

  15. Counterexample A counterexample is a single case in which a conjecture is not true.

  16. COUNTEREXAMPLE Example 8 On her first road trip, Little Window Watcher Wilma observes a number of vehicles. Each one she observes has four wheels. She conjectures “All vehicles have four wheels.” What is wrong with her conjecture? What counterexample will disprove it? Answer in your notebook Conjecture: All vehicles have 4 wheels

  17. Example 9 Kenny makes the following conjecture about the sum of two numbers. Find a counterexample to disprove Kenny’s conjecture. Conjecture: The sum of two numbers is always greater than the larger number. Answer in your notebook

  18. Example 10 Joe has a friend who just happens to be a Native American named Victor. One day Victor gave Joe a CD. The next day Victor decided that he wanted the CD back, and so he confronted Joe. After reluctantly giving the CD back to his friend, Joe made the conjecture: “Victor, like all Native Americans, is an Indian Giver.” What is wrong with his conjecture? What does this example illustrate? Answer in your notebook

  19. Inductive Fallacies The previous example illustrated an inductive fallacy, where a reliable conjecture cannot be justifiably made. Joe was guilty of a Hasty Generalization, basing a conclusion on too little information. Here are some others: • Unrepresentative Sample • False Analogy • Slothful Induction • Fallacy of Exclusion

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