Chapter 5 Indirect Proof How often have I

Chapter 5: Indirect Proof How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the true. Sherlock Holmes in the “Sign of the Four” by Sir Arthur Conan Doyle

Indirect Proof, New Postulate of Contradiction: If a proposition contradicts a true proposition, then it is false. Postulate of Elimination: If one of a given set of propositions must be true, and all but one of those propositions have been proved to be false, then this one remaining proposition must be true.

You already do this! Which is the capital of Mali? A. Paris B. Tuscon C. London D. Bamako Which Italian scientist used a new invention called the telescope to discover the moons of Jupiter? A. Sir Edmund Halley B. Julius Caesar C. Galileo Galilei D. Madonna

Indirect Proof Methodology 1. List all possibilities for the conclusion. 2. Negate the given conclusion. 3. Write a chain of reasons until you reach a contradiction: • contradiction of given information; or • contradiction of a theorem, a definition, or another known fact. 4. State the remaining possibility (from step 1) as the desired conclusion.

Example 1 C A D B

Example 2 B C A D

Example 3 H J K

Example 4 Given : AC ^ BD, BC @ EC, and AB @ ED A Prove : ÐB @ ÐCED E B C D

Indirect Proof, New Postulate of Contradiction: If a proposition contradicts a true proposition, then it is false. Postulate of Elimination: If one of a given set of propositions must be true, and all but one of those propositions have been proved to be false, then this one remaining proposition must be true.

Indirect Proof Methodology 1. List all possibilities for the conclusion. 2. Negate the given conclusion. 3. Write a chain of reasons until you reach a contradiction: • contradiction of given information; or • contradiction of a theorem, a definition, or another known fact. 4. State the remaining possibility (from step 1) as the desired conclusion.