Question 2 2.5 Indirect proofs are often confused with direct proofs of the contrapositive.are not the same, however, since in a direct proof of the contrapositive we assumthe negation of the conclusion while in an indirect proof we assume and use bohypothesis and the negation of the conclusion. Suppose we want to prove the theoimplies Q. A direct proof of the contrapositive would start with not Q and concluP. One way to formulate the argument would be to start by assuming P (the hypotand not Q (the RAA hypothesis), then to use the same proof as before to concluP and finally to end by saying that we must reject the RAA hypothesis because whave both P and not P, an obvious contradiction. While this is logically correcconsidered to be bad form and sloppy thinking; this way of organizing a proof sĩtherefore, usually be avoided.7An even worse misuse of proof by contradiction is the following. Suppose agarwe want to prove the theorem P implies Q. We assume the hypothesis P. Then wsuppose not Q (RAA hypothesis). After that we proceed to prove that P impliesthat point in the proof we have both Q and not Q. That is a contradiction, so wethe RAA hypothesis and conclude Q. In this case the structure of an indirect arghas been erected around a direct proof, thus obscuring the real proof. Again this isthinking. It is an abuse of indirect proof and should always be avoided.Use RAA proofs when they are helpful, but don't misuse them. A proof isdiscovered as an indirect proof because we can suppose the conclusion is false and ethe consequences. Once you have found a proof, you should reexamine it to seelogic can be simplified and the essence of the proof presented more directly.CISES 2.51. Identify the hypothesis and conclusion of each of the following statements.(a) If it rains, then I get wet.(b) If the sun shines, then we go hiking and biking.(c) If x > 0, then there exists a y such that y² = 0.(d) If 2x + 1 = 5, then either x2. State the converse and contrapositive of each of the statements in Exercise 1.3. Write the negation of each of the statements in Exercise 1.4. Write each of the following statements in "if..., then..." form.(a) It is necessary to score at least 90% on the test in order to receive an A.(b) A sufficient condition for passing the test is a score of 50% or higher.(c) You fail only if you score less than 50%.(d) You succeed whenever you try hard.5. State the converse and contrapositive of each of the statements in Exercise 4.6. Write the negation of each of the statements in Exercise 4.7. Restate each of the following assertions in "if..., then..." form.(a) Perpendicular lines must intersect.2 or x3.We will occasionally find good reason to formulate our proofs this way even thouggenerallygood practice.32 Chapter 2Axiomatic Systems and Incidence Geometry(b) Any two great circles on S² intersect.(c) Congruent triangles are similar.(d) Every triangle has angle sum less than or equal to 180°.8. Identify the hypothesis and conclusion of each of the following statements.(a) I can take topology if I pass geometry.(b) I get wet whenever it rains.(c) A number is divisible by 4 only if it is even.9. Restate using quantifiers.(a) Every triangle has an angle sum of 180°.(b) Some triangles have an angle sum of less than 180°.

1. Identify the hypothesis and conclusion of each of the following statements. (a) If it rains, then I get wet. (b) If the sun shines, then we go hiking and biking. (c) If x > 0, then there exists a y such that y = 0. (d) If 2x + 1 = 5, then either x = 2. State the converse and contrapositive of each of the statements in Exercise 1. 2 or x = 3.