1.1 Inductive and Deductive Reasoning 5 When forming a general conclusion using inductive reasoning, you should test it with several special cases to see whether the conclusion appears correct. If a special case is found that satisfies the conditions of the conjecture but produces a different result, such a case is called a counterexample . A counterexample proves that the conjecture is false because only one exception is needed to show that a conjecture is not valid. Galileo’s counterexample disproved Aristotle’s conjecture. If a counterexample cannot be found, the conjecture is neither proven nor disproven. Consider the statement “All birds fly.” A penguin is a bird that does not fly. Therefore, a penguin is a counterexample to the statement “All birds fly.” Deductive Reasoning A second type of reasoning process is called deductive reasoning , or deduction . Mathematicians use deductive reasoning to prove conjectures true or false. Definition: Deductive Reasoning Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. Example 5 illustrates deductive reasoning. General Specific Deductive Reasoning Inductive Reasoning Timely Tip The following diagram helps explain the difference between inductive reasoning and deductive reasoning. Inductive reasoning is the process of reasoning to a general conclusion through observations of specific cases. Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. In Example 4, we conjectured , using specific examples and inductive reasoning, that the result would be twice the original number selected. In Example 5, we proved , using deductive reasoning, that the result will always be twice the original number selected. Number Tricks Using Deductive Reasoning Instructor Resources for Section 1.1 in MyLab Math • Objective-Level Videos 1.1 • Animation: Number Tricks Using Deductive Reasoning • PowerPoint Lecture Slides 1.1 • MyLab Exercises and Assignments 1.1 Example 5 Use Deductive Reasoning to Prove a Conjecture Prove, using deductive reasoning, that the procedure given in Example 4 will always result in twice the original number selected. Solution To use deductive reasoning, we begin with the general case rather than specific examples. In Example 4, specific cases were used. Let’s select the letter n to represent any number . Pick any number: n Multiply the number by 4: n4 n4 means 4 times n. Add 2 to the product: n4 2 + Divide the sum by 2: + = + = + n n n 4 2 2 4 2 2 2 2 1 2 1 1 1 Subtract 1 from the quotient: n n 2 1 1 2 + − = Note that for any number n selected, the result is n2 , or twice the original number selected. Since n represented any number, we are beginning with the general case. Thus, this is deductive reasoning. 7 Now try Exercise 41
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