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Indirect Proofs
Objective
Use Indirect Reasoning to Write Proofs.
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Definition
So far, all proofs have used two laws of deductive reasoning—the Law of Detachment and the Law of Syllogism (see Section 2.5.) In this section, we learn a new type of reasoning called indirect reasoning. In indirect reasoning, all possibilities are considered and then all but one are proved false. The remaining possibility must be true. A proof involving indirect reasoning is an indirect proof.
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3. How to Write an Indirect Proof
Step 1. Assume that the opposite of the statement you are trying to prove is true. Step 2. Logically try to prove that the statement in Step 1 is true. HOWEVER, a contradiction should occur. Step 3. When we arrive at a contradiction, then the statement in Step 1 is false. Thus, the original statement must be true.
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4. Writing an Indirect Proof
Suppose you want to write an indirect proof of each statement. What would you assume? a. An integer n is divisible by 5. b. You do not have soccer practice today. Solution a. The opposite of “is divisible by” is “is not divisible by.” Assume temporarily that n is not divisible by 5.
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5. Writing an Indirect Proof
Suppose you want to write an indirect proof of each statement. As the second step of the proof, what would you assume? a. An integer n is divisible by 5. b. You do not have soccer practice today. Solution b. The opposite of “do not have” is “do have.” Assume temporarily that you do have soccer practice today.
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6. Identifying Contradictions
Which two statements contradict each other? I. II. III. Solution Statements I and II: Segments can be parallel and congruent. Thus, statements I and II do not contradict each other. Statements II and III: Segments can be congruent and perpendicular. Thus, statements II and III do not contradict each other.
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7. Identifying Contradictions
Which two statements contradict each other? I. II. III. Solution Statements I and III: Parallel segments do not intersect, so they cannot be perpendicular. Thus, statements I and III DO contradict each other.
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