Deductive and Inductive Reasoning, Study notes of Mathematics

Download Deductive and Inductive Reasoning and more Study notes Mathematics in PDF only on Docsity! The Nature of Mathematics Problem Solving and Reasoning Target Outcomes At the end of the lesson, you are expected to use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts: specifically, use Venn diagrams to verify the validity of a deductive argument and use inductive reasoning to establish a conjecture. Abstraction Even in todays’ world of digital revolution, there are still many unfamiliar situations arising that need solutions and answers. If we are faced with problems, puzzles, or dilemma, we try to reason out everything to arrive at solutions or answer. One of the many goals of teaching mathematics is to help provide solutions and find answers to these many unfamiliar situations that are a problem or inconvenience to our everyday life. George Polya is one of the most noted mathematicians known for his strategy in solving problems. In his book, “How To Solve It”, which sold over a million copies, he outlines the four-step strategy in problem solving- understand the problem, devise a plan, carry out the plan, and review the solution, to help students become a better problem solver. In this lesson, we will try to help you become a better problem solver by analyzing and exploring two types of logic or reasoning- deductive and inductive- and to try to demonstrate that problem solving can also be an enjoyable experience by solving problems involving patterns and “recreational” ones. Lesson 5. Deductive and Inductive Reasoning 5.1 DEDUCTIVE REASONING Deductive reasoning involves the application of general assumptions, procedures, or statements to a specific case after. According to Aristotle, a syllogism, an argument composed of two premises (major and minor) followed by all logical thoughts, is necessary for any inquiry. The best known of Aristotle’s syllogism is of the following general form, which is known as modus ponens: 1. If A, then B. All A are B. (major premise) 2. x is A. We have A. (minor premise) Therefore, x is B. Therefore, we have B. (conclusion) The validity of a deductive argument, like the above, can be shown using a Venn diagram. Using Venn diagram to illustrate most of the statements used in deductive argument, consider the following: If A, then B. (All A are B.) No A are B. Some A are B. (At least one A is B.) Example 1: Verify the validity of the following argument: 1. All professional wrestlers are actors. (major premise) 2. The Rock is an actor. (minor premise) Therefore, The Rock is a professional wrestler. (conclusion) Solution: Let us illustrate the major and minor premise using Venn diagram. We will have two cases as follows: Case 1: Case 2: In Case 1, the argument would appear to be valid because x = The Rock is placed inside the set of professional wrestlers, from the major premise, are actors. Hence, the conclusion “The Rock is a professional wrestler.” However, if x is placed outside the set of professional wrestlers, as in Case 2, it does not support the conclusion. Since the process of reasoning to obtain the conclusion is uncertain, the argument is invalid. Even though, in fact, The Rock is a professional wrestler. Note: If the argument is invalid, it does not mean the conclusion is false. That is, an invalid argument can have a true conclusion. In logic, validity and truth does not have the same meaning. Validity refers to the process of reasoning used to obtain a conclusion; truth refers to conformity with facts. If you can construct a Venn diagram in which the premises are met yet the conclusion does not necessarily follow, then an argument is invalid. Solution: Let n represent the number we picked. Multiply the number by 6: 6n Add 10 to the product: 6n + 10 Divide the sum by 2: (6n + 10) / 2 = 3n + 5 Subtract 5: (3n + 5) – 5 = 3n We started with n and ended with 3n. The procedure given in this example produces a number that is three times the original number. 5.2 INDUCTIVE REASONING Inductive reasoning involves generalizing a conclusion after examining specific assumptions, procedures, statements, and a pattern has been recognized and established. Mathematicians use both deductive and inductive reasoning to form conjectures. Some conjectures are formed by examining specific examples, looking for patterns, and making generalizations. The conclusion in an inductive argument is never guaranteed, although it may in fact be true, it may just also seem to follow a pattern. Now, we will try to establish conjectures using inductive reasoning. Example 1: Use inductive reasoning to predict the next number in each of the following lists. a. 5, 10, 15, 20, 25, ? b. 2, 5, 10, 17, 26, ? Solution: a. Each successive number is 5 larger than the preceding number. Thus, we predict that the most probable next number in the list is 5 larger than 25, which is 30. b. The first two numbers differ by 3. The second and the third numbers differ by 5. It appears that the difference between any two numbers is always 2 more than the preceding difference. Since 17 and 26 differ by 9, we predict that the next number in the list will be 11 larger than 26, which is 37. Example 2: Complete the procedure below for several different numbers and use inductive reasoning to establish a conjecture about the size of the resulting number and the size of the original number. a. Pick a number, b. multiply the number by 8, c. add 6 to the product, d. divide the sum by 2, and e. subtract 3 from the quotient. Solution: Suppose we pick 10. Multiply 10 by 8, we have 80. Adding 6 to 80, we have 86. Dividing 86 by 2, we have 43. Lastly, we subtracting 3 from 43, we get 40. Let’s try again another number, say 2. Multiply 2 by 8, we have 16. Adding 6 to 16, we have 22. Dividing 22 by 2, we have 11. Lastly, we subtract 3 from 11, we get 8. Now, you can pick different numbers, too. What can you induce if you follow the procedure? How is the resulting number compared to the original number you picked? The first number we picked is 10. After following the procedure, we get 40. If we compare the numbers, we can say that 40 is four times the original number, 10. The second number we picked is 2. After following the procedure, we get 8. If we again compare the numbers, we can say that 8 is four times the original number, 2. In each of the numbers we picked, it is conjectured that the procedure produces a number that is four times the original number. Example 3: Complete the procedure below for several different numbers and use inductive reasoning to establish a conjecture about the size of the resulting number and the size of the original number. a. Pick a number, b. multiply the number by 15, c. subtract 6 from the product, d. divide the difference by 3, and e. add 2. Solution: Suppose we pick 60. Multiply 60 by 15, we have 900. Subtracting 6 from 900, we have 894. Dividing 894 by 3, we have 298. Lastly, adding 2 to 298, we get 300. Let’s try again another number, say -5. Multiply -5 by 15, we have -75. Subtracting 6 from -75, we have -81. Dividing -81 by 3, we have -27. Lastly, adding 2 to -27, we get -25. Now, you can pick different numbers, too. What can you induce if you follow the procedure? How is the resulting number compared to the original number you picked? The first number we picked is 60. After following the procedure, we get 300. If we compare the numbers, we can say that 300 is five times the original number, 60. The second number we picked is -5. After following the procedure, we get -25. If we again compare the numbers, we can say that -25 is five times the original number, -5. In each of the numbers we picked, it is conjectured that the procedure produces a number that is five times the original number. Example 4: Use the data in the table and by inductive reasoning answer the following questions below. Earthquake Magnitude Maximum Tsunami Height (meters) 5.5 5 5.6 8 5.7 11 5.8 14 5.9 17 6.0 20 6.1 23 6.2 26 a. If the earthquake magnitude is 6.7, how high (in meters) will the tsunami be? b. Can a tsunami occur when the earthquake magnitude is less than 5? Explain your answer. Solution: a. From the given table above, every 0.1 increase in the earthquake magnitude, the maximum tsunami height increases by 3 meters. Hence, it is conjectured that the tsunami height for a 6.7-magnitude earthquake is at most 41 meters. b. No, because the maximum tsunami height for a 5.4- magnitude earthquake is 2 meters. Less than this magnitude, tsunami does not occur. Example 5: Galileo Galilei used inductive reasoning to discover that the time required for a pendulum to complete a swing, called the period of the pendulum, depends on the length of the pendulum. The following table shows some results obtained for pendulums of various lengths. For the sake of convenience, a length of 10 inches has been designated as 1 unit. Use the data in the table and inductive reasoning to answer each of the following: Length of pendulum, in units Period of pendulum, in heartbeats 1 1 4 2 9 3 16 4 a. If a pendulum has a length of 25 units, what is its period? b. If the length of the pendulum is quadrupled, what happens to its period?