AIM: WHAT IS AN INDIRECT PROOF?

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Presentation transcript:

AIM: WHAT IS AN INDIRECT PROOF? DO NOW: Define the term “indirect proof.” Math is yummy!

Remember: The “given” statements are always valid. INDIRECT PROOFS DEFINITION: An indirect proof is a logic proof that proves a statement true by proving that the contradictory statement is untrue. Given: Specific statements that are true. You must use the “given” statements to solve the proof. Remember: The “given” statements are always valid.

Model Problem Statement Reason A or B Both options Assume B One option C —> B Given C Detachment ~C ~B Contrapositive inference A Only other option Given: A or B C —› B Prove that A ˜ C is true A Start with both options Assume the incorrect option Use the given Prove the incorrect option untrue State that the other option is true

C —> B proves C true, but C is not true. This makes C —> B untrue, so B is untrue, as well. A is the only remaining option. You need to find a given statement that would be invalid if the assumed option was true. That given statement is evidence that the assumed option is incorrect.

EXAMPLE Bob studied or did not study. If Bob studied, Bob passed. If Bob did not study, he failed. Bob did not fail. Prove that Bob studied. SOLVE THIS PROBLEM

PROOF STATEMENT REASON 1. These are the two options. 1. Either: Bob studied or did not study. 1. These are the two options. 2. Assume: Bob did not study. 2. This is one possibility. 3. If Bob did not study, he failed. 3. Given 4. Bob failed. 4. Detachment. 5. But Bob did not fail. 5. Given. 6. So: The statement “Bob did not study” is false. 6. Contrapositive inference. 7. Bob studied. 7. This is the only remaining Option.

IMPORTANT CONCEPTS Statement: A sentence that can be determined to be true or false. Letters are used to represent statements (usually p and q). Conjunction ( ) = and. p q ~p ~q p ~q ~p q Disjunction ( ) = or. p q ~p ~q p ~q ~p q Conditional (—>) = implies. If, then statement. Ex., If you do your homework, I’ll take you to the movies.

Conditionals If today is Saturday, there is no school. p —>q a. inverse: (negate both) If today is not Saturday, there is school. ~p —>~q b. converse: (switch) If there is no school, today is Saturday. q —> p c. contrapositive: (negate and switch) If there is school, then, today is not Saturday. ~q —>~p Logically equivalent: same truth value. original statement = contrapositive inverse = converse

Laws of Logic Chain rule: p —>q q —> r p —> r Disjunctive inference: p q p q ~p ~q q p Contrapositive: p —> q ~ p —> ~ q Contrapositive inference: p—>q ~q ~ p DO YOUR MATH !