1. Inductive & Deductive Reasoning MATH 102 Contemporary Math S. Rook

2. Overview Section 1.2 in the textbook: – Inductive reasoning – Deductive reasoning

3. Inductive Reasoning

4. Inductive Reasoning: the process of drawing a conclusion by observing a pattern in specific instances. This conclusion is called a hypothesis or conjecture (guess) – i.e. going from specific examples to a general conclusion – e.g. A stray animal comes by your house four consecutive nights. You predict it will come on the fifth consecutive night – e.g. You’ve always gotten a birthday card from your grandmother so naturally you expect to get a birthday card this year from your grandmother

5. Inductive Reasoning (Continued) The key to inductive reasoning is the observation of patterns: – e.g. 1, 2, 3, 4, ?, ? – e.g. - 1 ⁄ 2, ¼, - 1 ⁄ 8, ?, ? Inductive reasoning is based on pattern recognition, NOT scientific fact. Thus, it is possible for an inductive hypothesis to be incorrect – See example 4 on pg 21 in the textbook – e.g. The stray animal does not come on the 5 th night – e.g. Your grandmother forgets to send you a birthday card this year

6. Inductive Reasoning (Example) Ex 1: Use inductive reasoning to solve. a) 8, -4, 2, ?, ?, ? b) The numbers 90, 264, and 8256 are all evenly divisible by 6. What can be noticed about all 3 numbers? Add the digits in each number, observe a pattern, and then form a conjecture

7. Deductive Reasoning

8. Deductive Reasoning: the process of using accepted [scientific] facts and general principles to arrive at a specific conclusion – i.e. going from general (known) to the specific – e.g. A medical examiner uses his knowledge of the temperature of the human body to determine the time of death of a murder victim – e.g. Solving the equation 3x – 1 = 5 for x Must feel comfortable in differentiating between inductive and deductive reasoning – See exercises 7 – 16 on pg 23 of the textbook

9. Inductive & Deductive Reasoning Ex 2: Consider the following process: (1) choose any natural (positive) number (2) multiply the number by 3 (3) add 9 to the product you just found (4) divide the result by 3 (5) subtract the number you started with a) Apply the process to three numbers. What are the results? b) Make a conjecture for part a). Does this involve inductive or deductive reasoning? c) Let n be the number you pick. Use the rules of Algebra to prove your conjecture. Does this involve inductive or deductive reasoning?

10. Summary After studying these slides, you should know how to do the following: – Reason inductively – Reason deductively – Understand the differences between inductive and deductive reasoning Additional Practice: – See the list of suggested problems for 1.2 Next Lesson: – Estimation (Section 1.3)