1. © 2008 Pearson Addison-Wesley. All rights reserved 1-1 Chapter 1 Section 1-1 Types of Reasoning

2. © 2008 Pearson Addison-Wesley. All rights reserved 1-2 Characteristics of Inductive and Deductive Reasoning Inductive Reasoning Use specific premise(info) to draw a general conclusion (a conjecture) from repeated observations. May or may not be true. Deductive Reasoning Apply general principles to draw specific conclusions. Absolute truth

3. © 2008 Pearson Addison-Wesley. All rights reserved 1-3 Determine whether the reasoning is an example of deductive or inductive reasoning. All math teachers have a great sense of humor. Patrick is a math teacher. Therefore, Patrick must have a great sense of humor. Solution Because the reasoning goes from general to specific, deductive reasoning was used. No other conclusion can be deduced. Example: determine the type of reasoning

4. © 2008 Pearson Addison-Wesley. All rights reserved 1-4 Determine whether the reasoning is an example of deductive or inductive reasoning. Carrie’s first three children were boys. If she has another child, it will be a boy. Solution Because the reasoning goes from specific to general, inductive reasoning was used. She may also have a girl. Example: determine the type of reasoning

5. © 2008 Pearson Addison-Wesley. All rights reserved 1-5 Use inductive reasoning to determine the probable next number in the list below. 1, 1, 2, 3, 5, 8, ____ (Fibonacci sequence) Use inductive reasoning to determine the probable next number in the list below. O, T, T, F, F, S, S, E, N, _____ Example: find next term

6. © 2008 Pearson Addison-Wesley. All rights reserved 1-6 Use inductive reasoning to determine the probable next number in the list below. 1, 3, 5, 7, ____ (arithmetic sequence) Use inductive reasoning to determine the probable next number in the list below. 2, 4, 8, 16, _____ (geometric sequence) Example: find next term

7. © 2008 Pearson Addison-Wesley. All rights reserved 1-7 Use the list of equations and inductive reasoning to predict the next multiplication fact in the list: 37 × 3 = 111 37 × 6 = 222 37 × 9 = 333 37 × 12 = 444 Solution 37 × 15 = 555 Example: predict the product of two numbers

8. © 2008 Pearson Addison-Wesley. All rights reserved 1-8 Use inductive reasoning to determine the probable next number in the list below. 2, 9, 16, 23, 30 Solution Each number in the list is obtained by adding 7 to the previous number. The probable next number is 30 + 7 = 37. Example: predicting the next number in a sequence

9. © 2008 Pearson Addison-Wesley. All rights reserved 1-9 Pitfalls of Inductive Reasoning One can not be sure about a conjecture until a general relationship has been proven. One counterexample is sufficient to make the conjecture false.

10. © 2008 Pearson Addison-Wesley. All rights reserved 1-10 We concluded that the probable next number in the list 2, 9, 16, 23, 30 is 37. If the list 2, 9, 16, 23, 30 actually represents the dates of Mondays in June, then the date of the Monday after June 30 is July 7 (see the figure on the next slide). The next number on the list would then be 7, not 37. Example: pitfalls of inductive reasoning

11. © 2008 Pearson Addison-Wesley. All rights reserved 1-11 Example: pitfalls of inductive reasoning

12. © 2008 Pearson Addison-Wesley. All rights reserved 1-12 Find the length of the hypotenuse in a right triangle with legs 3 and 4. Use the Pythagorean Theorem: c 2 = a 2 + b 2, where c is the hypotenuse and a and b are legs. Solution c 2 = 3 2 + 4 2 c 2 = 9 + 16 = 25 c = 5 (absolute truth – general to specific) Example: use deductive reasoning

13. © 2008 Pearson Addison-Wesley. All rights reserved 1-13 Find the sum 1+2+3+……+798+799+800. Solution 400 sums of 801 = 400(801)=320,400 Example: Use the method of Gauss

14. © 2008 Pearson Addison-Wesley. All rights reserved 1-14 Modify the method of Gauss to find the sum 2+4+6+……+100. Solution Factor out 2 first 2 times 25 sums of 51 =2(25)(51) Test question Example: Use the method of Gauss

15. © 2008 Pearson Addison-Wesley. All rights reserved 1-15 Modify the method of Gauss to find the sum 8+13+18+23+……43+48+53. Solution Note: the difference is 5 8+53=61 13+48=61 5(61)=305 Example: Use the method of Gauss