To prove that \(p \Rightarrow q\), we proceed as follows: Suppose \(p\Rightarrow q\) is false; that is, assume that \(p\) is true and \(q\) is false. When you get there, you are the only ones there. Rachel looks at you and says, ''If the art festival was today, there would be hundreds of people here, so it can't be today.'' Congruence 2. Discrete Math Basic Proof Methods 1.6 Introduction to Proofs Indirect Proof Example Theorem (For all integers n) If 3n+2 is odd, then n is odd. ¥Eventually arrive at some logical absurdity, e.g. The Euclidean Algorithm 4. Proof. two $\U_n$ 5. Argue until we There are two kinds of indirect proofs: the proof by contrapositive, and the proof by contradiction. Suppose you and your friend Rachel are going to an art festival. í¦úïûQifìñ¬½›6I³ˆÆ¢HŠ—HJ$şHè(�õB[#µÕb»»Pâ+ß\P�±îQÖ38î 0y®„ñZFã„Z’öâşJ\7fuQÍq9ØyŸÕÍ>­ÊrÇh'DR‹ò?ˆû1�¾H�ò’ ç²ÂUÄy÷í¼->ˆˆƒ¯³DkmH†èupí*q\üHÒú=3û(ú!Æ¢Æòrì'’6eòßÄI”JšÑşL¥öI%¶O The Chinese Remainder Theorem 8. Suppose that the conclusion is false, i.e., that n is even. Proof by Contradiction Another indirect proof is proof by contradiction. Then n = 2k for some Note the not . †‘&c„)ÇÁw).¾cÎ…~Àv™óÎàg¥Â?øB¥Åœ¢Û÷­¨8F:| ׂ¼Ì«Å¡aEÙÖúî44p«–Õ„…T¢ë1¦õ2“Ø:jάAF(²ø4®DY¯ÿ$=Â�. With an indirect proof, instead of proving that something must be true, you prove it indirectly by showing that it cannot be false. The proof by contrapositive is based on the fact that an implication is equivalent to its contrapositive. Therefore, instead of proving p \Rightarrow q, we may prove its contrapositive \overline {q} \Rightarrow \overline {p}. The Fundamental Theorem of Arithmetic 6. The GCD and the LCM 7. When your task in a proof is to prove that things are not congruent, not perpendicular, and so on, it’s a dead giveaway that you’re dealing with an indirect proof. You take out your tickets, look at the date and say, ''The date on the tickets is for tomorrow, so the art festival is not today.'' Indirect Proofs ¥Instead of starting with the given/known facts, we start by assuming the opposite of what we seek to prove. Notice that both you and Rachel came to the same conclusion, but you got to that concl… $\Z_n$ 3. Indirect Proof 3 Number Theory 1. ¥Use logical reasoning to deduce a sequence of facts.