Math 2 Geometry Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein 2.2 Indirect Proof

Statements Related to Conditional Conditional (or Implication) P → Q If P, then Q Converse of Conditional Q → P If Q, then P Inverse of Conditional ~P → ~Q If not P, then not Q Contrapositive of Conditional ~Q → ~P If not Q, then not P

If Tom lives in Ventura, he lives in California (P→Q) Converse (Q → P): If Tom lives in California, he lives in Ventura. Inverse (~P →~Q) If Tom doesn’t live in Ventura, he doesn’t live in California. Contrapositive (~Q → ~P) If Tom doesn’t live in California, he doesn’t live in Ventura. In general the conditional and contrapositive are either both true or both false.

If two angles are vertical angles, then they are congruent angles. Converse (Q → P): If two angles are congruent then they are vertical. Inverse (~P →~Q) If two angles are not vertical, then they are not congruent. Contrapositive (~Q → ~P) If two angles are not congruent, then they are not vertical.

Law of Detachment (From section 1.1) 1. If P, then Q 2. P C.  Q

Law of Negative Inference 1. P → Q 2. ~Q C.  ~P If I forget to water my flowers, they will wilt. My flowers did not wilt. Therefore, I did not forget to water them

Indirect Proof Law of Negative Inference is “backbone” Use when we want to prove something “isn’t” Proof in paragraph form Begin: “Suppose that….” or “Assume that…” Assume is negation of what we want to prove. We are finished when we find a contradiction

Method of Indirect Proof Given: P Prove: Q 1. Suppose that ~Q is true. 2. Reason from the supposition until you reach a contradiction. 3. Note that the supposition of ~Q is true must be false and that Q must therefore be true. [Completes the proof]

Example of Indirect Proof First need to do intro to section 2.3