Proofs 2.2 Method #2-Prove the Contrapositive Remember that an implication("P implies Q") is logically equivalent to its contrapos itive("not Q implies not P); proving one is as good as proving the other. And often proving the contrapositive is easier than proving the original statement. If so, then you can proceed as follows 1. Write, We prove the contrapositive: and then state the contrapositive 2. Proceed as in Method #1 Example Theorem 2. If r is irrational, then Vr is also irrational Recall that rational numbers are equal to a ratio of integers and irrational numbers are not. So we must show that if r is not a ratio of integers, then vr is also not a ratio of integers. That's pretty convoluted! We can eliminate both"not"s and make the proof straightforward by considering the contrapositive instead Proof. We prove the contrapositive: if Vr is rational, then r is rational Assume that Vr is rational. Then there exists integers a and b such that Squaring both sides give Since a2 and b2 are integers, r is also rational 3 A Bogus Technique: Reasoning Backward Somewhere out in America there must be dozens of high school teachers whispering into innocent ears that"reasoning backward"in proofs is fine.oh, yes, jusssst fine. Probably they sacrifice little furry animals on moonless nights, too. In any case, they re wrong Let's use the utterly incorrect, but depressingly popular technique of reasoning back ward to"prove"a famous inequalit Theorem 3(Arithmetic-Geometric Mean Inequality) For all nonnegative real numbers aProofs 7 2.2 Method #2 ­ Prove the Contrapositive Remember that an implication (“P implies Q”) is logically equivalent to its contrapos￾itive (“not Q implies not P”); proving one is as good as proving the other. And often proving the contrapositive is easier than proving the original statement. If so, then you can proceed as follows: 1. Write, “We prove the contrapositive:” and then state the contrapositive. 2. Proceed as in Method #1. Example Theorem 2. If r is irrational, then √r is also irrational. Recall that rational numbers are equal to a ratio of integers and irrational numbers are not. So we must show that if r is not a ratio of integers, then √r is also not a ratio of integers. That’s pretty convoluted! We can eliminate both “not”’s and make the proof straightforward by considering the contrapositive instead. Proof. We prove the contrapositive: if √r is rational, then r is rational. Assume that √r is rational. Then there exists integers a and b such that: √ a r = b Squaring both sides gives: 2 a r = b2 Since a2 and b2 are integers, r is also rational. 3 A Bogus Technique: Reasoning Backward Somewhere out in America there must be dozens of high school teachers whispering into innocent ears that “reasoning backward” in proofs is fine... oh, yes, jusssst fine. Probably they sacrifice little furry animals on moonless nights, too. In any case, they’re wrong. Let’s use the utterly incorrect, but depressingly popular technique of reasoning back￾ward to “prove” a famous inequality. Theorem 3 (Arithmetic­Geometric Mean Inequality). For all nonnegative real numbers a and b: a + b √ ab 2 ≥