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Negate each of the following statements. (a) There exists a model for incidence geometry in which the Postulate holds. (b) In every model for incidence geometry there are exactly sever (c) Every triangle has an angle sum of 180°. (d) Every triangle has an angle sum of less than 180°. (e) It is hot and humid outside. (f) My fovorito color is rod or groon

Negate each of the following statements. (a) There exists a model for incidence geometry in which the Postulate holds. (b) In every model for incidence geometry there are exactly sever (c) Every triangle has an angle sum of 180°. (d) Every triangle has an angle sum of less than 180°. (e) It is hot and humid outside. (f) My fovorito color is rod or groon

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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2.5 question 10

1:42 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB (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°. (c) Not every triangle has angle sum 180°. (d) Any two great circles on S2 intersect. (e) If P is a point and l is a line, then there is a line m such that P lies on m and m is perpendicular to l. 10. Negate each of the following statements. (a) There exists a model for incidence geometry in which the Euclidean Parallel Postulate holds. (b) In every model for incidence geometry there are exactly seven points. (c) Every triangle has an angle sum of 180°. (d) Every triangle has an angle sum of less than 180°. (e) It is hot and humid outside. (f) My favorite color is red or green. (g) If the sun shines, then we go hiking. (h) All geometry students know how to write proofs. 11. Negate each of the three parallel postulates stated in Section 2.3. 12. Construct truth tables that illustrate De Morgan's laws (page 26). 13. Construct a truth table which shows that the conditional statement H=C is logically equivalent to (not H) or C. Then use one of De Morgan's Laws to conclude that not (H = C) is logically equivalent to H and (not C). 14. Construct a truth table which shows directly that the negation of H = C is logically equivalent to H and (not C). 15. State the Pythagorean Theorem in "if..., then..." form. 16. Restate each of the three parallel postulates from Section 2.3 in "if..., then..." form. ME THEOREMS FROM INCIDENCE GEOMETRY Ve illustrate the lessons of the last section with several theorems and a proof from cidence geometry. The theorems in this section are theorems in incidence geometry, so eir proofs are to be based on the three incidence axioms that were stated in §2.2. One the hardest lessons to be learned in writing the proofs is that we may use only what is tually stated in the axioms, nothing more. Here, again, are the three axioms. cidence Axiom 1. For every pair of distinct points P and Q there exists exactly one line l ch that both P and Q lie on l. cidence Axiom 2. For every line l there exist at least two distinct points P and Q such at both P and Q lie on l. ncidence Axiom 3. There exist three points that do not all lie on any one line. The first theorem was already used as an example earlier in the chapter. As explained the last section, this theorem must be restated before it is ready for a proof. We will lopt the custom of formally restating theorems in if...then... form when necessary. efinition 2.6.1. Two lines are said to intersect if there exists a point that lies on both nes.
Transcribed Image Text:1:42 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB (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°. (c) Not every triangle has angle sum 180°. (d) Any two great circles on S2 intersect. (e) If P is a point and l is a line, then there is a line m such that P lies on m and m is perpendicular to l. 10. Negate each of the following statements. (a) There exists a model for incidence geometry in which the Euclidean Parallel Postulate holds. (b) In every model for incidence geometry there are exactly seven points. (c) Every triangle has an angle sum of 180°. (d) Every triangle has an angle sum of less than 180°. (e) It is hot and humid outside. (f) My favorite color is red or green. (g) If the sun shines, then we go hiking. (h) All geometry students know how to write proofs. 11. Negate each of the three parallel postulates stated in Section 2.3. 12. Construct truth tables that illustrate De Morgan's laws (page 26). 13. Construct a truth table which shows that the conditional statement H=C is logically equivalent to (not H) or C. Then use one of De Morgan's Laws to conclude that not (H = C) is logically equivalent to H and (not C). 14. Construct a truth table which shows directly that the negation of H = C is logically equivalent to H and (not C). 15. State the Pythagorean Theorem in "if..., then..." form. 16. Restate each of the three parallel postulates from Section 2.3 in "if..., then..." form. ME THEOREMS FROM INCIDENCE GEOMETRY Ve illustrate the lessons of the last section with several theorems and a proof from cidence geometry. The theorems in this section are theorems in incidence geometry, so eir proofs are to be based on the three incidence axioms that were stated in §2.2. One the hardest lessons to be learned in writing the proofs is that we may use only what is tually stated in the axioms, nothing more. Here, again, are the three axioms. cidence Axiom 1. For every pair of distinct points P and Q there exists exactly one line l ch that both P and Q lie on l. cidence Axiom 2. For every line l there exist at least two distinct points P and Q such at both P and Q lie on l. ncidence Axiom 3. There exist three points that do not all lie on any one line. The first theorem was already used as an example earlier in the chapter. As explained the last section, this theorem must be restated before it is ready for a proof. We will lopt the custom of formally restating theorems in if...then... form when necessary. efinition 2.6.1. Two lines are said to intersect if there exists a point that lies on both nes.
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