2.1.3 Using Deductive Reasoning to Verify Conjectures Key Objectives • Apply the Law of Detachment and the Law of Syllogism in logical reasoning. Key Terms • Deductive reasoning is the process of using logic to draw conclusions based on given facts, definitions, and properties. Theorems, Postulates, Corollaries, and Properties • Law of Detachment If p → q is a true statement and p is true, then q is true. • Law of Syllogism If p → q and q → r are true statements, then p → r is a true statement Proving that a conjecture is false can be simple because only one counterexample is required. The process for proving that a conjecture is true is more involved. Deductive reasoning, reasoning based on given facts, definitions, and properties, must be used to prove that a conjecture is true. The Laws of Detachment and Syllogism are forms of deductive reasoning that will be explained in this lesson. Example 1 Determining if a Conclusion is Based on Inductive or Deductive Reasoning Inductive reasoning is logic based on specific examples. Using a counterexample to disprove a conjecture is an example of inductive reasoning. Deductive reasoning is logic based on given facts, definitions, and properties. Conclusions made based on known definitions, research, or data are examples of deductive reasoning.

In this example, the conclusion is that the myth is false. This conclusion is based on observation of a specific example, opening a carbonated beverage after it is shaken and tapped, which is a counterexample to the given conditional. Therefore, the type of reasoning used is inductive reasoning.

In this example, the conclusion is that the myth is false, which was also the conclusion in the previous example. However, this conclusion is based on scientific research instead of on a specific example. Therefore, the type of reasoning used is deductive reasoning.

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