Plane Geometry - Bruce E. Shapiro

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22 SECTION 5. LOGIC AND PROOFRene Descartes explaining math to Queen Christina of Sweden (Pierre Louis Dumesnil,1698-1781); copy by Nils Forsberg (1884).We can then derive the following additional rules of logic using truth tables.∼ (∼ A) ⇐⇒ A (5.1)∼ (A ⇒ B) ⇐⇒ A∧ ∼ B (5.2)∼ A ⇒∼ B ⇐⇒ A ⇒ B (5.3)∼ (A ∧ B) ⇐⇒ (∼ A) ∨ (∼ B) (5.4)∼ [∀xF (x)] ⇐⇒ ∃x ∼ F (x) (5.5)∼ [∃xF (x)] ⇐⇒ ∀x ∼ F (x) (5.6)The following rule is known as the rule of detachment or modus ponens,[A ∧ (A ⇒ B)] ⇒ B (5.7)In other words, if A is true, and we know that A ⇒ B, then we also know« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 5. LOGIC AND PROOF 23that B is true. We also have the following rules of deduction:[(A ⇒ B) ∧ (B ⇒ C)] ⇒ [A ⇒ C] (5.8)[A ∧ B] ⇒ A (5.9)For the following example, recall that a rational number is any number xthat can be expressed as the ratio of two integers, e.g., x = p/q, where bothp and q are integers. An irrational number is a number that is not rational.Example 5.1 Prove that √ 2 is irrational.Proof.1. Assume that √ 2 is rational. (RAA hypothesis)2. There exists integers p and q such that √ 2 = p/q (Definition of arational number)3. Assume that p and q have no common factors (from our knowledgeof numbers we know that we can cancel out all common factors).4. Therefore at least one of p and q is odd. (otherwise there would be acommon factor of 2, which contradicts step 35. Since p 2 = 2q 2 , p is divisible by 2, hence it is even.6. Write p = 2s where s is an integer (definition of even).7. Then 4s 2 = p 2 = 2q 2 ⇒ q 2 = 2s 28. Then q 2 is divisible by 2, hence it is even (definition of even).9. Hence q is even because the square of any odd number must be odd.10. Since p and q are even this contradicts step 4. Hence our RAA assumptionis false.We numbered the steps in this proof for clarity. In general we don’t do this,because it is somewhat tedious and takes up a lot of space, and insteadusually write them as a paragraph. However, until you are comfortablewriting proofs, you should write them out in a step-by-step manner.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
Page 1: Foundations of GeometryLecture Note
Page 4 and 5: ivCONTENTS30 Triangles in Neutral G
Page 6 and 7: 2 SECTION 1. PREFACEare enrolled in
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Page 16 and 17: 12 SECTION 3. CA STANDARDIn 2005 Ca
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74 SECTION 14. BETWEENNESS« CC BY-
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76 SECTION 15. THE PLANE SEPARATION
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78 SECTION 15. THE PLANE SEPARATION
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80 SECTION 16. ANGLESFigure 16.1: T
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90 SECTION 17. THE CROSSBAR THEOREM
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92 SECTION 18. LINEAR PAIRSFigure 1
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94 SECTION 18. LINEAR PAIRSγ + ∠
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96 SECTION 19. ANGLE BISECTORSwith
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98 SECTION 19. ANGLE BISECTORSAngle
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100 SECTION 20. THE CONTINUITY AXIO
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102 SECTION 20. THE CONTINUITY AXIO
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104 SECTION 21. SIDE-ANGLE-SIDEBirk
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106 SECTION 21. SIDE-ANGLE-SIDEFigu
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108 SECTION 22. NEUTRAL GEOMETRYEll
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110 SECTION 22. NEUTRAL GEOMETRY«
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112 SECTION 23. ANGLE-SIDE-ANGLEFig
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116 SECTION 24. EXTERIOR ANGLESTheo
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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122 SECTION 26. SIDE-SIDE-SIDECase
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124 SECTION 26. SIDE-SIDE-SIDEBy si
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126 SECTION 27. SCALENE AND TRIANGL
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128 SECTION 27. SCALENE AND TRIANGL
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130 SECTION 28. CHARACTERIZATION OF
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132 SECTION 28. CHARACTERIZATION OF
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134 SECTION 29. TRANSVERSALSFigure
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136 SECTION 29. TRANSVERSALSCorolla
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138 SECTION 30. TRIANGLES IN NEUTRA
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168 SECTION 33. RECTANGLESLemma 33.
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188 SECTION 37. AREADefinition 37.5
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238 SECTION 43. INDIANA BILL 246the
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240 SECTION 44. ESTIMATING πFigure
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246 SECTION 44. ESTIMATING πTable
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252 SECTION 45. EUCLIDEAN CONSTRUCT
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264 SECTION 46. HYPERBOLIC GEOMETRY
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272SECTION 47.PERPENDICULAR LINES I
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274SECTION 47.PERPENDICULAR LINES I
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306 SECTION 52. ARC LENGTHAs we see
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310 SECTION 52. ARC LENGTHThus the
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312 SECTION 52. ARC LENGTHMeasuring
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314 SECTION 53. SPHERICAL GEOMETRYW
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322 APPENDIX A. SYMBOLS USEDTechnol
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Plane Geometry - Bruce E. Shapiro

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