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Section 1.1 Inductive and Deductive Reasoning 1. 2. Inductive Reasoning Objectives: Understand and use inductive reasoning. Understand and use deductive reasoning. The process of arriving at a general conclusion based on observations of specific examples. Definitions: Conjecture/hypothesis: The conclusion formed as a result of inductive reasoning which may or may not be true. Counterexample: A case for which the conjecture is not true which proves the conjecture is false. 1 Strong Inductive Argument 2 Weak Inductive Argument This cooler contains 30 soda cans I looked at 27 of the cans, and they were all Coke Probably all 30 of the cans are Coke Conjecture? all 30 sodas are Coke Counterexample? if you pull out a soda that is not Coke Since neither my dad nor my brother ever cried in front of me, men have difficulty expressing their feelings. This conclusion is based on just two observations. This sample is neither random nor large enough to represent all men. 3 Example 1 4 Example 1 - solution Identifying a pattern in a list of numbers using addition Since the numbers are increasing relatively slowly, try addition. What number comes next? 3, 8, 13, 18, ___ 3, 8, +5 5 13, +5 18, +5 ___ +5 The common difference between each pair of numbers is 5. Therefore, the next number is 18 + 5 = 23. 6 1 Example 2 Example 2 - Solution Identifying a pattern in a list of numbers using multiplication Since the numbers are increasing relatively quickly, try multiplication. 3, x3 What number comes next? 3, 9, 27, 81, ___ 9, 27, x3 81, x3 ___ x3 The common ratio between each pair of numbers is 3. Thus, the next number is: 81 x 3 = 243. 7 Example 3 Fibonacci Sequence 8 Example 4 - Predicting the next figure in a sequence by finding the pattern What comes next in this list of numbers? 1, 1, 2, 3, 5, 8, 13, 21, ? , Solution: This pattern is formed by adding the previous 2 numbers to get the next number: So the next number in the sequence is: 13 + 21 = 34 , , ... What is the next figure in the sequence? Solution: 9 Example 5 – find the next figure in the pattern , , 10 Inductive Reasoning: More than one Solution! Is this illustration a wine goblet or two faces looking at each other? ,... Solution: 11 12 2 Inductive Reasoning: More than one Solution! Deductive Reasoning 2, 4, ? What is the next number in this sequence? The process of drawing a specific conclusion from one or more general statements. A theorem is a conclusion proved true by deductive reasoning If the pattern is to add 2 to the previous number it is 6. If the pattern is to multiply the previous number by 2 then the answer is 8. We need to know one more number to decide. 13 Example 5 An Example in Everyday Life Everyday Situation The ad says “All Jewelry On Sale - 50% off”. You want to buy a ring. You only pay $50 for a ring normally priced at $100. 14 Example 6 Deductive Reasoning Using Inductive Reasoning, apply the rules to specific numbers. Do you see a pattern? General Statement: All jewelry is 50% off. A ring is jewelry. Conclusion: A ring originally $100 is sale priced at $50. Select any number 4 7 11 4 +3 = 7 7 + 3 = 10 11 +3 = 14 Multiply by 4 7 x 4 = 28 10 x 4 = 40 14 x 4 = 56 Divide this product by 2 28 = 14 2 14 – 6 = 8 40 = 20 2 56 = 28 2 28 – 6 = 22 Add 3 Subtract 6 from the quotient 15 20 – 6 = 14 The output number is twice the input number 16 Example 6 continued Using Deductive reasoning: Select any number Add 3 Multiply by 4 Divide by 2 Subtract 6 n n+3 (n + 3) x 4 (n + 3) × 4 4n + 12 = = 2n + 6 2 2 2n + 6 – 6 = 2n Which agrees with our inductive hypothesis. 17 3